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Theorem dfdom2 8188
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2 ≼ = ( ≺ ∪ ≈ )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 8165 . . 3 ≺ = ( ≼ ∖ ≈ )
21uneq2i 3928 . 2 ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ ))
3 uncom 3921 . 2 ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ )
4 enssdom 8187 . . 3 ≈ ⊆ ≼
5 undif 4211 . . 3 ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ )
64, 5mpbi 221 . 2 ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼
72, 3, 63eqtr3ri 2796 1 ≼ = ( ≺ ∪ ≈ )
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cdif 3731  cun 3732  wss 3734  cen 8159  cdom 8160  csdm 8161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-opab 4874  df-xp 5285  df-rel 5286  df-f1o 6077  df-en 8163  df-dom 8164  df-sdom 8165
This theorem is referenced by:  brdom2  8192
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