Theorem dtru 5435

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 2031. The same comments and revision history concerning axiom usage as in exneq 5434 apply. (Contributed by NM, 7-Nov-2006.) Extract exneq 5434 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 5434	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnal 1829	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbi 229	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1539  ∃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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782

This theorem is referenced by:  brprcneu  6878  zfcndpow  10607