Theorem epelg 4275

Description: The epsilon relation and membership are the same. General version of epel 4277. (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 3990	. . . 4 ⊢ (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2		elopab 4243	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦))
3		vex 2733	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 2733	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5	3, 4	pm3.2i 270	. . . . . . . . . 10 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
6		opeqex 4234	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	5, 6	mpbiri 167	. . . . . . . . 9 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
8	7	simpld 111	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
9	8	adantr 274	. . . . . . 7 ⊢ ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
10	9	exlimivv 1889	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥 ∈ 𝑦) → 𝐴 ∈ V)
11	2, 10	sylbi 120	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦} → 𝐴 ∈ V)
12		df-eprel 4274	. . . . 5 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
13	11, 12	eleq2s 2265	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
14	1, 13	sylbi 120	. . 3 ⊢ (𝐴 E 𝐵 → 𝐴 ∈ V)
15	14	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 → 𝐴 ∈ V))
16		elex 2741	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
17	16	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 → 𝐴 ∈ V))
18		eleq12 2235	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝑦 ↔ 𝐴 ∈ 𝐵))
19	18, 12	brabga 4249	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
20	19	expcom 115	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)))
21	15, 17, 20	pm5.21ndd 700	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586   class class class wbr 3989  {copab 4049   E cep 4272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274

This theorem is referenced by:  epelc  4276  efrirr  4338  smoiso  6281  ecidg  6577  ordiso2  7012  ltpiord  7281