Theorem epelg 4380

Description: The epsilon relation and membership are the same. General version of epel 4382. (Contributed by Scott Fenton, 27-Mar-2011.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
epelg	⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Proof of Theorem epelg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4083	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		elopab 4345	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦))
3		vex 2802	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 2802	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 272	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6		opeqex 4335	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	5, 6	mpbiri 168	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	simpld 112	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
9	8	adantr 276	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
10	9	exlimivv 1943	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
11	2, 10	sylbi 121	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V)
12		df-eprel 4379	. . . . 5 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
13	11, 12	eleq2s 2324	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
14	1, 13	sylbi 121	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
15	14	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
16		elex 2811	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
17	16	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
18		eleq12 2294	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
19	18, 12	brabga 4351	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
20	19	expcom 116	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
21	15, 17, 20	pm5.21ndd 710	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   class class class wbr 4082  {copab 4143   E cep 4377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-eprel 4379

This theorem is referenced by:  epelc  4381  efrirr  4443  smoiso  6446  ecidg  6744  ordiso2  7198  ltpiord  7502