Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-opelidb1ALT Structured version   Visualization version   GIF version

Theorem bj-opelidb1ALT 34553
 Description: Characterization of the couples in I. (Contributed by BJ, 29-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-opelidb1ALT (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-opelidb1ALT
StepHypRef Expression
1 df-br 5054 . . 3 (𝐴 I 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ I )
2 reli 5686 . . . 4 Rel I
32brrelex1i 5596 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3sylbir 238 . 2 (⟨𝐴, 𝐵⟩ ∈ I → 𝐴 ∈ V)
5 inex1g 5210 . . 3 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
6 bj-opelid 34543 . . 3 ((𝐴𝐵) ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
75, 6syl 17 . 2 (𝐴 ∈ V → (⟨𝐴, 𝐵⟩ ∈ I ↔ 𝐴 = 𝐵))
84, 7biadanii 821 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480   ∩ cin 3918  ⟨cop 4556   class class class wbr 5053   I cid 5447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator