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Mirrors > Home > MPE Home > Th. List > relsdom | Structured version Visualization version GIF version |
Description: Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.) |
Ref | Expression |
---|---|
relsdom | ⊢ Rel ≺ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldom 8990 | . 2 ⊢ Rel ≼ | |
2 | reldif 5828 | . . 3 ⊢ (Rel ≼ → Rel ( ≼ ∖ ≈ )) | |
3 | df-sdom 8987 | . . . 4 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
4 | 3 | releqi 5790 | . . 3 ⊢ (Rel ≺ ↔ Rel ( ≼ ∖ ≈ )) |
5 | 2, 4 | sylibr 234 | . 2 ⊢ (Rel ≼ → Rel ≺ ) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ Rel ≺ |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3960 Rel wrel 5694 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-ss 3980 df-opab 5211 df-xp 5695 df-rel 5696 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: domdifsn 9093 sucdom2OLD 9121 sdom0OLD 9148 sdomirr 9153 sdomdif 9164 sucdom2 9241 0sdom1dom 9272 sdom1OLD 9277 1sdom2dom 9281 unxpdom 9287 unxpdom2 9288 sucxpdom 9289 isfinite2 9332 fin2inf 9340 fodomfir 9366 card2on 9592 djuxpdom 10224 djufi 10225 infdif 10246 cfslb2n 10306 isfin5 10337 isfin6 10338 isfin4p1 10353 fin56 10431 fin67 10433 sdomsdomcard 10598 gchi 10662 canthp1lem1 10690 canthp1lem2 10691 canthp1 10692 frgpnabl 19908 fphpd 42804 sdomne0 43403 sdomne0d 43404 |
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