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Theorem unissel 3765
 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 3764 . . 3 (𝐵𝐴𝐵 𝐴)
32adantl 275 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐵 𝐴)
41, 3eqssd 3114 1 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480   ⊆ wss 3071  ∪ cuni 3736 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737 This theorem is referenced by:  elpwuni  3902
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