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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 8529 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | sdomdomtr 8643 | . 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶) | |
3 | 1, 2 | sylan2 594 | 1 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 class class class wbr 5059 ≈ cen 8499 ≼ cdom 8500 ≺ csdm 8501 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 |
This theorem is referenced by: sdomen2 8655 unxpdom2 8719 sucxpdom 8720 findcard3 8754 fofinf1o 8792 sdomsdomcardi 9393 cardsdomel 9396 cardmin2 9420 alephnbtwn2 9491 pwsdompw 9619 infdif2 9625 fin23lem27 9743 axcclem 9872 numthcor 9909 sdomsdomcard 9975 pwcfsdom 9998 cfpwsdom 9999 inawinalem 10104 inatsk 10193 r1tskina 10197 tskuni 10198 rucALT 15578 iunmbl2 24153 dirith2 26102 erdszelem10 32468 mblfinlem1 34964 pellex 39508 rp-isfinite6 39958 harval3 39978 |
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