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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 8929 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
2 | sdomnsym 9049 | . . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
3 | sdomnen 8928 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
4 | ensym 8950 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
5 | 3, 4 | nsyl3 138 | . . 3 ⊢ (𝐴 ≈ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
6 | 2, 5 | jaoi 856 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → ¬ 𝐵 ≺ 𝐴) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 846 class class class wbr 5110 ≈ cen 8887 ≼ cdom 8888 ≺ csdm 8889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 |
This theorem is referenced by: sdom0OLD 9060 sdomdomtr 9061 domsdomtr 9063 sdomdif 9076 onsdominel 9077 nndomogOLD 9177 sdom1OLD 9194 fofinf1o 9278 carddom2 9920 fidomtri 9936 fidomtri2 9937 infxpenlem 9956 alephordi 10017 infdif 10152 infdif2 10153 cfslbn 10210 cfslb2n 10211 fincssdom 10266 fin45 10335 domtriom 10386 alephval2 10515 alephreg 10525 pwcfsdom 10526 cfpwsdom 10527 pwfseqlem3 10603 gchpwdom 10613 gchaleph 10614 hargch 10616 gchhar 10622 winainflem 10636 rankcf 10720 tskcard 10724 vdwlem12 16871 odinf 19352 rectbntr0 24211 erdszelem10 33834 finminlem 34819 fphpd 41168 |
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