Theorem nd4 9749

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)

Assertion

Ref	Expression
nd4	⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥)

Proof of Theorem nd4

Step	Hyp	Ref	Expression
1		nd3 9748	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑧 𝑦 ∈ 𝑥)
2	1	aecoms 2394	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-reg 8788

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-un 3797  df-nul 4142  df-sn 4399  df-pr 4401

This theorem is referenced by:  axrepnd  9753