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Mirrors > Home > MPE Home > Th. List > domentr | Structured version Visualization version GIF version |
Description: Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
Ref | Expression |
---|---|
domentr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 7868 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
2 | domtr 7895 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan2 490 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 class class class wbr 4583 ≈ cen 7838 ≼ cdom 7839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-f1o 5811 df-en 7842 df-dom 7843 |
This theorem is referenced by: domdifsn 7928 xpdom1g 7942 domunsncan 7945 sdomdomtr 7978 domen2 7988 mapdom2 8016 php 8029 unxpdom2 8053 sucxpdom 8054 xpfir 8067 fodomfi 8124 cardsdomelir 8682 infxpenlem 8719 xpct 8722 infpwfien 8768 inffien 8769 mappwen 8818 iunfictbso 8820 cdaxpdom 8894 cdainflem 8896 cdainf 8897 cdalepw 8901 ficardun2 8908 unctb 8910 infcdaabs 8911 infunabs 8912 infcda 8913 infdif 8914 infxpdom 8916 pwcdadom 8921 infmap2 8923 fictb 8950 cfslb 8971 fin1a2lem11 9115 unirnfdomd 9268 iunctb 9275 alephreg 9283 cfpwsdom 9285 gchdomtri 9330 canthp1lem1 9353 pwfseqlem5 9364 pwxpndom 9367 gchcdaidm 9369 gchxpidm 9370 gchpwdom 9371 gchhar 9380 inttsk 9475 inar1 9476 tskcard 9482 znnen 14780 qnnen 14781 rpnnen 14795 rexpen 14796 aleph1irr 14814 cygctb 18116 1stcfb 21058 2ndcredom 21063 2ndcctbss 21068 hauspwdom 21114 tx1stc 21263 tx2ndc 21264 met1stc 22136 met2ndci 22137 re2ndc 22412 opnreen 22442 ovolctb2 23067 ovolfi 23069 uniiccdif 23152 dyadmbl 23174 opnmblALT 23177 vitali 23188 mbfimaopnlem 23228 mbfsup 23237 aannenlem3 23889 fnct 28876 dmvlsiga 29519 sigapildsys 29552 omssubadd 29689 carsgclctunlem3 29709 finminlem 31482 phpreu 32563 lindsdom 32573 mblfinlem1 32616 pellexlem4 36414 pellexlem5 36415 nnfoctb 38238 ioonct 38611 subsaliuncl 39252 caragenunicl 39414 aacllem 42356 |
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