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Theorem unissel 4962
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 482 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
2 elssuni 4961 . . 3 (𝐵𝐴𝐵 𝐴)
32adantl 481 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐵 𝐴)
41, 3eqssd 4026 1 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wss 3976   cuni 4931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-uni 4932
This theorem is referenced by:  elpwuni  5128  mretopd  23121  toponmre  23122  neiptopuni  23159  filunibas  23910  unidmvol  25595  unicls  33849  carsguni  34273  onintunirab  43188
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