No, the warlock's Pact Magic feature is a completely separate feature from the Mystic Arcanum feature, and each independently describes how you get spells without relying on or referring to the other. Pact Magic says you get another spell known at e.g. 11th level, so you do, period. The Mystic Arcanum feature, separately, gives you access to a new spell that you can use with the Mystic Arcanum feature.
If that reasoning is not convincing, here's a bit of circumstantial evidence underlining the necessary separation of the two features: Notice that the spells you gain access to with Pact Magic and Mystic Arcanum cannot be the same due to incompatible spell levels! All Mystic Arcanum spells you gain are of a level (6th and up) that is inaccessible to your Pact Magic, which tops out at 5th-level spells. For example, at 11th level, you gain a new spell known of maximum 5th-level via Pact Magic, and a 6th-level Mystic Arcanum spell. These cannot be the same since they are different spell levels.