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Theorem dfdom2 8139
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2 ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 8116 . . 3 ≺ = ( ≼ ∖ ≈ )
21uneq2i 3915 . 2 ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3 uncom 3908 . 2 ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4 enssdom 8138 . . 3 ≈ ⊆ ≼
5 undif 4192 . . 3 ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
64, 5mpbi 220 . 2 ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
72, 3, 63eqtr3ri 2802 1 ≼ = ( ≺ ∪ ≈ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  cdif 3720  cun 3721  wss 3723  cen 8110  cdom 8111  csdm 8112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-opab 4848  df-xp 5256  df-rel 5257  df-f1o 6037  df-en 8114  df-dom 8115  df-sdom 8116
This theorem is referenced by:  brdom2  8143
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