Theorem dtru 5377

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 2029. The same comments and revision history concerning axiom usage as in exneq 5376 apply. See dtruALT 5324 and dtruALT2 5306 for alternate proofs avoiding ax-pr 5368. (Contributed by NM, 7-Nov-2006.) Extract exneq 5376 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 5376	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnal 1828	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbi 230	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2120  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781

This theorem is referenced by:  brprcneu  6807  zfcndpow  10499