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Theorem dfdom2 7932
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2 ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 7909 . . 3 ≺ = ( ≼ ∖ ≈ )
21uneq2i 3747 . 2 ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3 uncom 3740 . 2 ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4 enssdom 7931 . . 3 ≈ ⊆ ≼
5 undif 4026 . . 3 ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
64, 5mpbi 220 . 2 ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
72, 3, 63eqtr3ri 2652 1 ≼ = ( ≺ ∪ ≈ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cdif 3556  cun 3557  wss 3559  cen 7903  cdom 7904  csdm 7905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-opab 4679  df-xp 5085  df-rel 5086  df-f1o 5859  df-en 7907  df-dom 7908  df-sdom 7909
This theorem is referenced by:  brdom2  7936
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