Theorem nd2 9594

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 8658	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2482	. . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦)
3	1	nfnth 1869	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ2 2145	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2537	. . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 208	. . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧)
7	1, 6	mto 188	. 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦
8		axc11 2448	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦))
9	7, 8	mtoi 190	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1622  [wsb 2038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047  ax-reg 8654

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-v 3334  df-dif 3710  df-un 3712  df-nul 4051  df-sn 4314  df-pr 4316

This theorem is referenced by:  axrepnd  9600  axpownd  9607  axinfndlem1  9611  axacndlem4  9616