Theorem distel 35804

Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5442 and elirrv 9636.) (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 5442	. . 3 ⊢ ∃𝑧 𝑥 ∈ 𝑧
2		df-ex 1780	. . . 4 ⊢ (∃𝑧 𝑥 ∈ 𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥 ∈ 𝑧)
3		nfnae 2439	. . . . . 6 ⊢ Ⅎ𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4		dveel1 2466	. . . . . . . 8 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝑧 → ∀𝑦 𝑥 ∈ 𝑧))
5	3, 4	nf5d 2284	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 ∈ 𝑧)
6	5	nfnd 1858	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥 ∈ 𝑧)
7		elequ2 2123	. . . . . . . 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 2412	. . . . 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 9636	. . . . 5 ⊢ ¬ 𝑦 ∈ 𝑦
15		elequ1 2115	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
16	14, 15	mtbii 326	. . . 4 ⊢ (𝑦 = 𝑥 → ¬ 𝑥 ∈ 𝑦)
17	16	alimi 1811	. . 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 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-reg 9632

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629

This theorem is referenced by: (None)