Theorem distel 35845

Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5378 and elirrv 9483.) (Contributed by Scott Fenton, 15-Dec-2010.)

Assertion

Ref	Expression
distel	⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)

Proof of Theorem distel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el 5378	. . 3 ⊢ ∃𝑧 𝑥 ∈ 𝑧
2		df-ex 1781	. . . 4 ⊢ (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧)
3		nfnae 2434	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		dveel1 2461	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
5	3, 4	nf5d 2286	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝑧)
6	5	nfnd 1859	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥 ∈ 𝑧)
7		elequ2 2126	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
8	7	notbid 318	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦))
9	8	a1i 11	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥 ∈ 𝑧 ↔ ¬ 𝑥 ∈ 𝑦)))
10	3, 6, 9	cbvald 2407	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
11	10	notbid 318	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
12	2, 11	bitrid 283	. . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦))
13	1, 12	mpbii 233	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
14		elirrv 9483	. . . . 5 ⊢ ¬ 𝑦 ∈ 𝑦
15		elequ1 2118	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
16	14, 15	mtbii 326	. . . 4 ⊢ (𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝑦)
17	16	alimi 1812	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥 ∈ 𝑦)
18	17	con3i 154	. 2 ⊢ (¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
19	13, 18	impbii 209	1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-sep 5232  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)