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| Mirrors > Home > MPE Home > Th. List > endomtr | Structured version Visualization version GIF version | ||
| Description: Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.) |
| Ref | Expression |
|---|---|
| endomtr | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | endom 9019 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
| 2 | domtr 9047 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
| 3 | 1, 2 | sylan 580 | 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 |
| 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-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: cnvct 9074 undomOLD 9100 xpdom1g 9109 xpdom3 9110 domunsncan 9112 sucdom2OLD 9122 domsdomtr 9152 domen1 9159 mapdom1 9182 mapdom2 9188 mapdom3 9189 phpOLD 9259 onomeneqOLD 9266 hartogslem1 9582 harcard 10018 infxpenlem 10053 infpwfien 10102 alephsucdom 10119 mappwen 10152 dfac12lem2 10185 djulepw 10233 fictb 10284 cfflb 10299 canthp1lem1 10692 pwfseqlem5 10703 pwxpndom2 10705 pwdjundom 10707 gchxpidm 10709 gchhar 10719 tskinf 10809 inar1 10815 gruina 10858 rexpen 16264 mreexdomd 17692 hauspwdom 23509 rectbntr0 24854 rabfodom 32524 snct 32725 dya2iocct 34282 finminlem 36319 iccioo01 37328 pibt2 37418 lindsdom 37621 poimirlem26 37653 heiborlem3 37820 pellexlem4 42843 pellexlem5 42844 safesnsupfidom1o 43430 sn1dom 43539 mpct 45206 thincciso2 49104 aacllem 49320 |
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