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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 8582 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8608 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
3 | 1, 2 | sylan 583 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 class class class wbr 5030 ≈ cen 8552 ≼ cdom 8553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-f1o 6346 df-en 8556 df-dom 8557 |
This theorem is referenced by: cnvct 8633 undom 8654 xpdom1g 8663 xpdom3 8664 domunsncan 8666 sucdom2 8676 domsdomtr 8702 domen1 8709 mapdom1 8732 mapdom2 8738 mapdom3 8739 php 8751 onomeneq 8788 hartogslem1 9079 harcard 9480 infxpenlem 9513 infpwfien 9562 alephsucdom 9579 mappwen 9612 dfac12lem2 9644 djulepw 9692 fictb 9745 cfflb 9759 canthp1lem1 10152 pwfseqlem5 10163 pwxpndom2 10165 pwdjundom 10167 gchxpidm 10169 gchhar 10179 tskinf 10269 inar1 10275 gruina 10318 rexpen 15673 mreexdomd 17023 hauspwdom 22252 rectbntr0 23584 rabfodom 30425 snct 30623 dya2iocct 31817 finminlem 34145 iccioo01 35118 pibt2 35211 lindsdom 35394 poimirlem26 35426 heiborlem3 35594 pellexlem4 40226 pellexlem5 40227 sn1dom 40687 mpct 42279 aacllem 45958 |
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