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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 8524 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domtr 8550 | . 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 5057 ≈ cen 8494 ≼ cdom 8495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-f1o 6355 df-en 8498 df-dom 8499 |
This theorem is referenced by: cnvct 8574 undom 8593 xpdom1g 8602 xpdom3 8603 domunsncan 8605 domsdomtr 8640 domen1 8647 mapdom1 8670 mapdom2 8676 mapdom3 8677 php 8689 onomeneq 8696 sucdom2 8702 hartogslem1 8994 harcard 9395 infxpenlem 9427 infpwfien 9476 alephsucdom 9493 mappwen 9526 dfac12lem2 9558 djulepw 9606 fictb 9655 cfflb 9669 canthp1lem1 10062 pwfseqlem5 10073 pwxpndom2 10075 pwdjundom 10077 gchxpidm 10079 gchhar 10089 tskinf 10179 inar1 10185 gruina 10228 rexpen 15569 mreexdomd 16908 hauspwdom 22037 rectbntr0 23367 rabfodom 30193 snct 30375 dya2iocct 31437 finminlem 33563 pibt2 34580 lindsdom 34767 poimirlem26 34799 heiborlem3 34972 pellexlem4 39307 pellexlem5 39308 sn1dom 39770 mpct 41340 aacllem 44830 |
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