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Mirrors > Home > MPE Home > Th. List > dfsdom2 | Structured version Visualization version GIF version |
Description: Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
dfsdom2 | ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sdom 8629 | . 2 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | sbthcl 8768 | . . 3 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | |
3 | 2 | difeq2i 4034 | . 2 ⊢ ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) |
4 | difin 4176 | . 2 ⊢ ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) = ( ≼ ∖ ◡ ≼ ) | |
5 | 1, 3, 4 | 3eqtri 2769 | 1 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∖ cdif 3863 ∩ cin 3865 ◡ccnv 5550 ≈ cen 8623 ≼ cdom 8624 ≺ csdm 8625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 |
This theorem is referenced by: brsdom2 8770 |
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