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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 8024 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8050 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan 487 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 class class class wbr 4685 ≈ cen 7994 ≼ cdom 7995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-f1o 5933 df-en 7998 df-dom 7999 |
This theorem is referenced by: cnvct 8074 undom 8089 xpdom1g 8098 xpdom3 8099 domunsncan 8101 domsdomtr 8136 domen1 8143 mapdom1 8166 mapdom2 8172 mapdom3 8173 php 8185 onomeneq 8191 sucdom2 8197 hartogslem1 8488 harcard 8842 infxpenlem 8874 infpwfien 8923 alephsucdom 8940 mappwen 8973 dfac12lem2 9004 cdalepw 9056 fictb 9105 cfflb 9119 canthp1lem1 9512 pwfseqlem5 9523 pwxpndom2 9525 pwcdandom 9527 gchxpidm 9529 gchhar 9539 tskinf 9629 inar1 9635 gruina 9678 rexpen 15001 mreexdomd 16357 hauspwdom 21352 rectbntr0 22682 rabfodom 29470 snct 29619 dya2iocct 30470 finminlem 32437 lindsdom 33533 poimirlem26 33565 heiborlem3 33742 pellexlem4 37713 pellexlem5 37714 mpct 39707 aacllem 42875 |
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