Theorem nd2 9745

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)

Assertion

Ref	Expression
nd2	⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Proof of Theorem nd2

Step	Hyp	Ref	Expression
1		elirrv 8790	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2428	. . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦)
3	1	nfnth 1846	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ2 2121	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2484	. . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 210	. . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧)
7	1, 6	mto 189	. 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦
8		axc11 2396	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦))
9	7, 8	mtoi 191	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1599  [wsb 2011

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 5017  ax-nul 5025  ax-pr 5138  ax-reg 8786

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  9751  axpownd  9758  axinfndlem1  9762  axacndlem4  9767