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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 7847 | . 2 ⊢ Rel ≼ | |
2 | reldif 5161 | . . 3 ⊢ (Rel ≼ → Rel ( ≼ ∖ ≈ )) | |
3 | df-sdom 7844 | . . . 4 ⊢ ≺ = ( ≼ ∖ ≈ ) | |
4 | 3 | releqi 5125 | . . 3 ⊢ (Rel ≺ ↔ Rel ( ≼ ∖ ≈ )) |
5 | 2, 4 | sylibr 223 | . 2 ⊢ (Rel ≼ → Rel ≺ ) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ Rel ≺ |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3537 Rel wrel 5043 ≈ 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 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-dom 7843 df-sdom 7844 |
This theorem is referenced by: domdifsn 7928 sdom0 7977 sdomirr 7982 sdomdif 7993 sucdom2 8041 sdom1 8045 unxpdom 8052 unxpdom2 8053 sucxpdom 8054 isfinite2 8103 fin2inf 8108 card2on 8342 cdaxpdom 8894 cdafi 8895 cfslb2n 8973 isfin5 9004 isfin6 9005 isfin4-3 9020 fin56 9098 fin67 9100 sdomsdomcard 9261 gchi 9325 canthp1lem1 9353 canthp1lem2 9354 canthp1 9355 frgpnabl 18101 fphpd 36398 |
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