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| Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) | 
| Ref | Expression | 
|---|---|
| dfdom2 | ⊢ ≼ = ( ≺ ∪ ≈ ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-sdom 8989 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
| 2 | 1 | uneq2i 4164 | . 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ )) | 
| 3 | uncom 4157 | . 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ ) | |
| 4 | enssdom 9018 | . . 3 ⊢ ≈ ⊆ ≼ | |
| 5 | undif 4481 | . . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ ) | |
| 6 | 4, 5 | mpbi 230 | . 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ | 
| 7 | 2, 3, 6 | 3eqtr3ri 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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