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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 8148 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8174 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan 489 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 class class class wbr 4804 ≈ cen 8118 ≼ cdom 8119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-f1o 6056 df-en 8122 df-dom 8123 |
This theorem is referenced by: cnvct 8198 undom 8213 xpdom1g 8222 xpdom3 8223 domunsncan 8225 domsdomtr 8260 domen1 8267 mapdom1 8290 mapdom2 8296 mapdom3 8297 php 8309 onomeneq 8315 sucdom2 8321 hartogslem1 8612 harcard 8994 infxpenlem 9026 infpwfien 9075 alephsucdom 9092 mappwen 9125 dfac12lem2 9158 cdalepw 9210 fictb 9259 cfflb 9273 canthp1lem1 9666 pwfseqlem5 9677 pwxpndom2 9679 pwcdandom 9681 gchxpidm 9683 gchhar 9693 tskinf 9783 inar1 9789 gruina 9832 rexpen 15156 mreexdomd 16511 hauspwdom 21506 rectbntr0 22836 rabfodom 29651 snct 29800 dya2iocct 30651 finminlem 32618 lindsdom 33716 poimirlem26 33748 heiborlem3 33925 pellexlem4 37898 pellexlem5 37899 mpct 39892 aacllem 43060 |
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