Theorem epelg 4027

Description: The epsilon relation and membership are the same. General version of epel 4029. (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 3765	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		elopab 3995	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦))
3		vex 2560	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 2560	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 257	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6		opeqex 3986	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	5, 6	mpbiri 157	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	simpld 105	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
9	8	adantr 261	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
10	9	exlimivv 1776	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
11	2, 10	sylbi 114	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V)
12		df-eprel 4026	. . . . 5 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
13	11, 12	eleq2s 2132	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
14	1, 13	sylbi 114	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
15	14	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
16		elex 2566	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
17	16	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
18		eleq12 2102	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
19	18, 12	brabga 4001	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
20	19	expcom 109	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
21	15, 17, 20	pm5.21ndd 621	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378   class class class wbr 3764  {copab 3817   E cep 4024

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-eprel 4026

This theorem is referenced by:  epelc  4028  efrirr  4090  smoiso  5917  ecidg  6170  ordiso2  6357  ltpiord  6417