Theorem nd2 10561

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nd2

Step	Hyp	Ref	Expression
1		elirrv 9547	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
2		stdpc4 2101	. . . 4 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → [𝑧 / 𝑦]𝑧 ∈ 𝑦)
3	1	nfnth 1825	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑧
4		elequ2 2160	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧))
5	3, 4	sbie 2536	. . . 4 ⊢ ([𝑧 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑧)
6	2, 5	sylib 221	. . 3 ⊢ (∀𝑦 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑧)
7	1, 6	mto 200	. 2 ⊢ ¬ ∀𝑦 𝑧 ∈ 𝑦
8		axc11 2464	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑧 ∈ 𝑦 → ∀𝑦 𝑧 ∈ 𝑦))
9	7, 8	mtoi 202	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561  [wsb 2093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2215  ax-13 2406  ax-sep 5251  ax-reg 9542

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807  df-sb 2094

This theorem is referenced by:  axrepnd  10567  axpownd  10574  axinfndlem1  10578  axacndlem4  10583