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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 8360 | . 2 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
2 | sbthcl 8486 | . . 3 ⊢ ≈ = ( ≼ ∩ ◡ ≼ ) | |
3 | 2 | difeq2i 4017 | . 2 ⊢ ( ≼ ∖ ≈ ) = ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) |
4 | difin 4158 | . 2 ⊢ ( ≼ ∖ ( ≼ ∩ ◡ ≼ )) = ( ≼ ∖ ◡ ≼ ) | |
5 | 1, 3, 4 | 3eqtri 2823 | 1 ⊢ ≺ = ( ≼ ∖ ◡ ≼ ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∖ cdif 3856 ∩ cin 3858 ◡ccnv 5442 ≈ cen 8354 ≼ cdom 8355 ≺ csdm 8356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 |
This theorem is referenced by: brsdom2 8488 |
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