Theorem dtru 5409

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 2063. The same comments and revision history concerning axiom usage as in exneq 5408 apply. See dtruALT 5354 and dtruALT2 5336 for alternate proofs avoiding ax-pr 5395. (Contributed by NM, 7-Nov-2006.) Extract exneq 5408 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 5408	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnal 1860	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	mpbi 233	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813

This theorem is used by:  brprcneu  6898  zfcndpow  10728