Theorem dtru 5387

Description: Given any set (the "𝑦 " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 2030. The same comments and revision history concerning axiom usage as in exneq 5386 apply. See dtruALT 5334 and dtruALT2 5316 for alternate proofs avoiding ax-pr 5378. (Contributed by NM, 7-Nov-2006.) Extract exneq 5386 as an intermediate result. (Revised by BJ, 2-Jan-2025.)

Assertion

Ref	Expression
dtru	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		exneq 5386	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnal 1829	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbi 230	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782

This theorem is referenced by:  brprcneu  6825  zfcndpow  10531