Theorem nd1 10166

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
nd1	⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)

Proof of Theorem nd1

Step	Hyp	Ref	Expression
1		elirrv 9190	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2076	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → [𝑧 / 𝑦]𝑦 ∈ 𝑧)
3	1	nfnth 1810	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ1 2119	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2505	. . . 4 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 221	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑧 → 𝑧 ∈ 𝑧)
7	1, 6	mto 200	. 2 ⊢ ¬ ∀𝑦 𝑦 ∈ 𝑧
8		axc11 2429	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 ∈ 𝑧 → ∀𝑦 𝑦 ∈ 𝑧))
9	7, 8	mtoi 202	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541  [wsb 2072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-13 2371  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-reg 9186

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-sn 4528  df-pr 4530

This theorem is referenced by:  axrepnd  10173  axinfndlem1  10184  axinfnd  10185  axacndlem1  10186  axacndlem2  10187