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| Mirrors > Home > MPE Home > Th. List > sdomentr | Structured version Visualization version GIF version | ||
| Description: Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.) |
| Ref | Expression |
|---|---|
| sdomentr | ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≺ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | endom 9019 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
| 2 | sdomdomtr 9150 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) | |
| 3 | 1, 2 | sylan2 593 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≺ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 |
| This theorem is referenced by: sdomen2 9162 unxpdom2 9290 sucxpdom 9291 findcard3OLD 9319 fofinf1o 9372 sdomsdomcardi 10011 cardsdomel 10014 cardmin2 10039 alephnbtwn2 10112 pwsdompw 10243 infdif2 10249 fin23lem27 10368 axcclem 10497 numthcor 10534 sdomsdomcard 10600 pwcfsdom 10623 cfpwsdom 10624 inawinalem 10729 inatsk 10818 r1tskina 10822 tskuni 10823 rucALT 16266 iunmbl2 25592 dirith2 27572 erdszelem10 35205 mblfinlem1 37664 pellex 42846 rp-isfinite6 43531 harval3 43551 |
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