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Mirrors > Home > MPE Home > Th. List > sdomdom | Structured version Visualization version GIF version |
Description: Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.) |
Ref | Expression |
---|---|
sdomdom | ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsdom 7864 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 class class class wbr 4583 ≈ 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-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-br 4584 df-sdom 7844 |
This theorem is referenced by: domdifsn 7928 sdomnsym 7970 sdomdomtr 7978 domsdomtr 7980 sdomtr 7983 sucdom2 8041 sucxpdom 8054 isfinite2 8103 pwfi 8144 card2on 8342 fict 8433 fidomtri2 8703 prdom2 8712 infxpenlem 8719 indcardi 8747 alephnbtwn2 8778 alephsucdom 8785 alephdom 8787 dfac13 8847 cdalepw 8901 infcdaabs 8911 infdif 8914 infunsdom1 8918 infunsdom 8919 infxp 8920 cfslb2n 8973 sdom2en01 9007 isfin32i 9070 fin34 9095 fin67 9100 hsmexlem1 9131 hsmex3 9139 entri3 9260 unirnfdomd 9268 alephexp1 9280 cfpwsdom 9285 gchdomtri 9330 canthp1 9355 pwfseqlem5 9364 gchcdaidm 9369 gchxpidm 9370 gchpwdom 9371 hargch 9374 gchaclem 9379 gchhar 9380 gchac 9382 inawinalem 9390 inar1 9476 rankcf 9478 tskuni 9484 grothac 9531 rpnnen 14795 rexpen 14796 aleph1irr 14814 dis1stc 21112 hauspwdom 21114 ovolfi 23069 sibfof 29729 heiborlem3 32782 harinf 36619 saluncl 39213 meadjun 39355 meaiunlelem 39361 omeunle 39406 |
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