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| 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 9023 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
| 2 | sdomnsym 9139 | . . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
| 3 | sdomnen 9022 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
| 4 | ensym 9044 | . . . 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 5142 ≈ cen 8983 ≼ cdom 8984 ≺ csdm 8985 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 | 
| This theorem is referenced by: sdom0OLD 9150 sdomdomtr 9151 domsdomtr 9153 sdomdif 9166 onsdominel 9167 nndomogOLD 9264 sdom1OLD 9280 fofinf1o 9373 carddom2 10018 fidomtri 10034 fidomtri2 10035 infxpenlem 10054 alephordi 10115 infdif 10249 infdif2 10250 cfslbn 10308 cfslb2n 10309 fincssdom 10364 fin45 10433 domtriom 10484 alephval2 10613 alephreg 10623 pwcfsdom 10624 cfpwsdom 10625 pwfseqlem3 10701 gchpwdom 10711 gchaleph 10712 hargch 10714 gchhar 10720 winainflem 10734 rankcf 10818 tskcard 10822 vdwlem12 17031 odinf 19582 rectbntr0 24855 erdszelem10 35206 finminlem 36320 fimgmcyc 42549 fphpd 42832 | 
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