Theorem epelg 5134

Description: The epsilon relation and membership are the same. General version of epel 5136. (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 4761	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		elopab 5087	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦))
3		vex 3307	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3307	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 470	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6		opeqex 5066	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	5, 6	mpbiri 248	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	simpld 477	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
9	8	adantr 472	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
10	9	exlimivv 1973	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
11	2, 10	sylbi 207	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V)
12		df-eprel 5133	. . . . 5 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
13	11, 12	eleq2s 2821	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
14	1, 13	sylbi 207	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
15	14	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
16		elex 3316	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
17	16	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
18		eleq12 2793	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
19	18, 12	brabga 5093	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
20	19	expcom 450	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
21	15, 17, 20	pm5.21ndd 368	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596  ∃wex 1817   ∈ wcel 2103  Vcvv 3304  ⟨cop 4291   class class class wbr 4760  {copab 4820   E cep 5132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-eprel 5133

This theorem is referenced by:  epelc  5135  efrirr  5199  efrn2lp  5200  predep  5819  epne3  7097  cnfcomlem  8709  fpwwe2lem6  9570  ltpiord  9822  orvcelval  30760  brcnvep  34272