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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 8978 | . 2 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
2 | sdomnsym 9098 | . . 3 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐵 ≺ 𝐴) | |
3 | sdomnen 8977 | . . . 4 ⊢ (𝐵 ≺ 𝐴 → ¬ 𝐵 ≈ 𝐴) | |
4 | ensym 8999 | . . . 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 5149 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 |
This theorem is referenced by: sdom0OLD 9109 sdomdomtr 9110 domsdomtr 9112 sdomdif 9125 onsdominel 9126 nndomogOLD 9226 sdom1OLD 9243 fofinf1o 9327 carddom2 9972 fidomtri 9988 fidomtri2 9989 infxpenlem 10008 alephordi 10069 infdif 10204 infdif2 10205 cfslbn 10262 cfslb2n 10263 fincssdom 10318 fin45 10387 domtriom 10438 alephval2 10567 alephreg 10577 pwcfsdom 10578 cfpwsdom 10579 pwfseqlem3 10655 gchpwdom 10665 gchaleph 10666 hargch 10668 gchhar 10674 winainflem 10688 rankcf 10772 tskcard 10776 vdwlem12 16925 odinf 19431 rectbntr0 24348 erdszelem10 34191 finminlem 35203 fphpd 41554 |
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