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Theorem dfdom2 8974
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2 ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 8942 . . 3 ≺ = ( ≼ ∖ ≈ )
21uneq2i 4161 . 2 ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3 uncom 4154 . 2 ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4 enssdom 8973 . . 3 ≈ ⊆ ≼
5 undif 4482 . . 3 ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
64, 5mpbi 229 . 2 ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
72, 3, 63eqtr3ri 2770 1 ≼ = ( ≺ ∪ ≈ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cdif 3946  cun 3947  wss 3949  cen 8936  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-xp 5683  df-rel 5684  df-f1o 6551  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  brdom2  8978
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