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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 8538 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | sdomdomtr 8652 | . 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 5068 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 |
This theorem is referenced by: sdomen2 8664 unxpdom2 8728 sucxpdom 8729 findcard3 8763 fofinf1o 8801 sdomsdomcardi 9402 cardsdomel 9405 cardmin2 9429 alephnbtwn2 9500 pwsdompw 9628 infdif2 9634 fin23lem27 9752 axcclem 9881 numthcor 9918 sdomsdomcard 9984 pwcfsdom 10007 cfpwsdom 10008 inawinalem 10113 inatsk 10202 r1tskina 10206 tskuni 10207 rucALT 15585 iunmbl2 24160 dirith2 26106 erdszelem10 32449 mblfinlem1 34931 pellex 39439 rp-isfinite6 39891 harval3 39911 |
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