Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-elid5 Structured version   Visualization version   GIF version

Theorem bj-elid5 34499
 Description: Characterization of the elements of I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid5 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))

Proof of Theorem bj-elid5
StepHypRef Expression
1 reli 5686 . . . 4 Rel I
2 df-rel 5550 . . . 4 (Rel I ↔ I ⊆ (V × V))
31, 2mpbi 233 . . 3 I ⊆ (V × V)
43sseli 3949 . 2 (𝐴 ∈ I → 𝐴 ∈ (V × V))
5 bj-elid4 34498 . 2 (𝐴 ∈ (V × V) → (𝐴 ∈ I ↔ (1st𝐴) = (2nd𝐴)))
64, 5biadanii 821 1 (𝐴 ∈ I ↔ (𝐴 ∈ (V × V) ∧ (1st𝐴) = (2nd𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480   ⊆ wss 3919   I cid 5447   × cxp 5541  Rel wrel 5548  ‘cfv 6344  1st c1st 7679  2nd c2nd 7680 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-pow 5254  ax-pr 5318  ax-un 7452 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-sbc 3759  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-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fv 6352  df-1st 7681  df-2nd 7682 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator