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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 8767 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8793 | . 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 396 class class class wbr 5074 ≈ cen 8730 ≼ cdom 8731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-f1o 6440 df-en 8734 df-dom 8735 |
This theorem is referenced by: cnvct 8824 undomOLD 8847 xpdom1g 8856 xpdom3 8857 domunsncan 8859 sucdom2OLD 8869 domsdomtr 8899 domen1 8906 mapdom1 8929 mapdom2 8935 mapdom3 8936 phpOLD 9005 onomeneqOLD 9012 hartogslem1 9301 harcard 9736 infxpenlem 9769 infpwfien 9818 alephsucdom 9835 mappwen 9868 dfac12lem2 9900 djulepw 9948 fictb 10001 cfflb 10015 canthp1lem1 10408 pwfseqlem5 10419 pwxpndom2 10421 pwdjundom 10423 gchxpidm 10425 gchhar 10435 tskinf 10525 inar1 10531 gruina 10574 rexpen 15937 mreexdomd 17358 hauspwdom 22652 rectbntr0 23995 rabfodom 30851 snct 31048 dya2iocct 32247 finminlem 34507 iccioo01 35498 pibt2 35588 lindsdom 35771 poimirlem26 35803 heiborlem3 35971 pellexlem4 40654 pellexlem5 40655 sn1dom 41133 mpct 42741 aacllem 46505 |
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