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Mirrors > Home > MPE Home > Th. List > dfdom2 | Structured version Visualization version GIF version |
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) |
Ref | Expression |
---|---|
dfdom2 | ⊢ ≼ = ( ≺ ∪ ≈ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sdom 7844 | . . 3 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | 1 | uneq2i 3726 | . 2 ⊢ ( ≈ ∪ ≺ ) = ( ≈ ∪ ( ≼ ∖ ≈ )) |
3 | uncom 3719 | . 2 ⊢ ( ≈ ∪ ≺ ) = ( ≺ ∪ ≈ ) | |
4 | enssdom 7866 | . . 3 ⊢ ≈ ⊆ ≼ | |
5 | undif 4001 | . . 3 ⊢ ( ≈ ⊆ ≼ ↔ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ ) | |
6 | 4, 5 | mpbi 219 | . 2 ⊢ ( ≈ ∪ ( ≼ ∖ ≈ )) = ≼ |
7 | 2, 3, 6 | 3eqtr3ri 2641 | 1 ⊢ ≼ = ( ≺ ∪ ≈ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 df-rel 5045 df-f1o 5811 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: brdom2 7871 |
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