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Mirrors > Home > MPE Home > Th. List > domnsym | Structured version Visualization version GIF version |
Description: Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) |
Ref | Expression |
---|---|
domnsym | ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdom2 8636 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
2 | sdomnsym 8749 | . . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
3 | sdomnen 8635 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
4 | ensym 8655 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
5 | 3, 4 | nsyl3 140 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
6 | 2, 5 | jaoi 857 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
7 | 1, 6 | sylbi 220 | 1 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 class class class wbr 5039 ≈ cen 8601 ≼ cdom 8602 ≺ csdm 8603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 |
This theorem is referenced by: sdom0 8756 sdomdomtr 8757 domsdomtr 8759 sdomdif 8772 onsdominel 8773 nndomog 8815 sdom1 8854 fofinf1o 8929 carddom2 9558 fidomtri 9574 fidomtri2 9575 infxpenlem 9592 alephordi 9653 infdif 9788 infdif2 9789 cfslbn 9846 cfslb2n 9847 fincssdom 9902 fin45 9971 domtriom 10022 alephval2 10151 alephreg 10161 pwcfsdom 10162 cfpwsdom 10163 pwfseqlem3 10239 gchpwdom 10249 gchaleph 10250 hargch 10252 gchhar 10258 winainflem 10272 rankcf 10356 tskcard 10360 vdwlem12 16508 odinf 18908 rectbntr0 23683 erdszelem10 32829 finminlem 34193 fphpd 40282 |
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