Theorem dtru 5432

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 2024. The same comments and revision history concerning axiom usage as in exneq 5431 apply. (Contributed by NM, 7-Nov-2006.) Extract exneq 5431 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 5431	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnal 1822	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbi 229	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1532  ∃wex 1774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ex 1775

This theorem is referenced by:  brprcneu  6881  zfcndpow  10633