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Theorem unissel 4899
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 483 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
2 elssuni 4898 . . 3 (𝐵𝐴𝐵 𝐴)
32adantl 482 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐵 𝐴)
41, 3eqssd 3961 1 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wss 3910   cuni 4865
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-in 3917  df-ss 3927  df-uni 4866
This theorem is referenced by:  elpwuni  5065  mretopd  22441  toponmre  22442  neiptopuni  22479  filunibas  23230  unidmvol  24903  unicls  32424  carsguni  32848  onintunirab  41538
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