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Theorem dfdom2 9019
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2 ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 8989 . . 3 ≺ = ( ≼ ∖ ≈ )
21uneq2i 4164 . 2 ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3 uncom 4157 . 2 ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4 enssdom 9018 . . 3 ≈ ⊆ ≼
5 undif 4481 . . 3 ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
64, 5mpbi 230 . 2 ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
72, 3, 63eqtr3ri 2773 1 ≼ = ( ≺ ∪ ≈ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cdif 3947  cun 3948  wss 3950  cen 8983  cdom 8984  csdm 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-rel 5691  df-f1o 6567  df-en 8987  df-dom 8988  df-sdom 8989
This theorem is referenced by:  brdom2  9023
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