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Theorem unissel 4398
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 471 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
2 elssuni 4397 . . 3 (𝐵𝐴𝐵 𝐴)
32adantl 480 . 2 (( 𝐴𝐵𝐵𝐴) → 𝐵 𝐴)
41, 3eqssd 3584 1 (( 𝐴𝐵𝐵𝐴) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wss 3539   cuni 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367
This theorem is referenced by:  elpwuni  4543  mretopd  20654  toponmre  20655  neiptopuni  20692  filunibas  21443  unidmvol  23061  unicls  29111  carsguni  29531
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