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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 9021 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
2 | sdomnsym 9137 | . . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
3 | sdomnen 9020 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
4 | ensym 9042 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
5 | 3, 4 | nsyl3 138 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
6 | 2, 5 | jaoi 857 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 847 class class class wbr 5148 ≈ 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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 |
This theorem is referenced by: sdom0OLD 9148 sdomdomtr 9149 domsdomtr 9151 sdomdif 9164 onsdominel 9165 nndomogOLD 9261 sdom1OLD 9277 fofinf1o 9370 carddom2 10015 fidomtri 10031 fidomtri2 10032 infxpenlem 10051 alephordi 10112 infdif 10246 infdif2 10247 cfslbn 10305 cfslb2n 10306 fincssdom 10361 fin45 10430 domtriom 10481 alephval2 10610 alephreg 10620 pwcfsdom 10621 cfpwsdom 10622 pwfseqlem3 10698 gchpwdom 10708 gchaleph 10709 hargch 10711 gchhar 10717 winainflem 10731 rankcf 10815 tskcard 10819 vdwlem12 17026 odinf 19596 rectbntr0 24868 erdszelem10 35185 finminlem 36301 fimgmcyc 42521 fphpd 42804 |
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