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Mirrors > Home > MPE Home > Th. List > sdomtr | Structured version Visualization version GIF version |
Description: Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.) |
Ref | Expression |
---|---|
sdomtr | ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 8221 | . 2 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domsdomtr 8335 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | |
3 | 1, 2 | sylan 576 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 class class class wbr 4841 ≼ cdom 8191 ≺ csdm 8192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 |
This theorem is referenced by: sdomn2lp 8339 2pwuninel 8355 2pwne 8356 r1sdom 8885 alephordi 9181 pwsdompw 9312 gruina 9926 rexpen 15289 |
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