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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 8882 | . 2 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | sbthcl 9035 | . . 3 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | |
3 | 2 | difeq2i 4077 | . 2 ⊢ ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) |
4 | difin 4219 | . 2 ⊢ ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) = ( ≼ ∖ ◡ ≼ ) | |
5 | 1, 3, 4 | 3eqtri 2768 | 1 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∖ cdif 3905 ∩ cin 3907 ◡ccnv 5630 ≈ cen 8876 ≼ cdom 8877 ≺ csdm 8878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 |
This theorem is referenced by: brsdom2 9037 |
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