ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  epelg GIF version

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 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))

Proof of Theorem epelg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 3765 . . . 4 (𝐴 E 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ E )
2 elopab 3995 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦))
3 vex 2560 . . . . . . . . . . 11 𝑥 ∈ V
4 vex 2560 . . . . . . . . . . 11 𝑦 ∈ V
53, 4pm3.2i 257 . . . . . . . . . 10 (𝑥 ∈ V ∧ 𝑦 ∈ V)
6 opeqex 3986 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
75, 6mpbiri 157 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87simpld 105 . . . . . . . 8 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → 𝐴 ∈ V)
98adantr 261 . . . . . . 7 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
109exlimivv 1776 . . . . . 6 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑦) → 𝐴 ∈ V)
112, 10sylbi 114 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} → 𝐴 ∈ V)
12 df-eprel 4026 . . . . 5 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
1311, 12eleq2s 2132 . . . 4 (⟨𝐴, 𝐵⟩ ∈ E → 𝐴 ∈ V)
141, 13sylbi 114 . . 3 (𝐴 E 𝐵𝐴 ∈ V)
1514a1i 9 . 2 (𝐵𝑉 → (𝐴 E 𝐵𝐴 ∈ V))
16 elex 2566 . . 3 (𝐴𝐵𝐴 ∈ V)
1716a1i 9 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ V))
18 eleq12 2102 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦𝐴𝐵))
1918, 12brabga 4001 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 E 𝐵𝐴𝐵))
2019expcom 109 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴 E 𝐵𝐴𝐵)))
2115, 17, 20pm5.21ndd 621 1 (𝐵𝑉 → (𝐴 E 𝐵𝐴𝐵))
Colors of variables: wff set class
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  6355  ltpiord  6415
  Copyright terms: Public domain W3C validator