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

Theorem unissel 3801
Description: Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
unissel (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)

Proof of Theorem unissel
StepHypRef Expression
1 simpl 108 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
2 elssuni 3800 . . 3 (𝐵𝐴𝐵 𝐴)
32adantl 275 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐵 𝐴)
41, 3eqssd 3145 1 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wss 3102   cuni 3772
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773
This theorem is referenced by:  elpwuni  3938
  Copyright terms: Public domain W3C validator