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Mirrors > Home > MPE Home > Th. List > ensdomtr | Structured version Visualization version GIF version |
Description: Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ensdomtr | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 9039 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domsdomtr 9178 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | |
3 | 1, 2 | sylan 579 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 class class class wbr 5166 ≈ cen 9000 ≼ cdom 9001 ≺ csdm 9002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 |
This theorem is referenced by: sdomen1 9187 sucxpdom 9318 f1finf1oOLD 9334 findcard3OLD 9347 isfinite2 9362 pm54.43 10070 infxpenlem 10082 alephnbtwn2 10141 alephordi 10143 alephsucdom 10148 pwsdompw 10272 infunsdom1 10281 cflim2 10332 fin23lem27 10397 cfpwsdom 10653 inawinalem 10758 inar1 10844 tskcard 10850 tskuni 10852 rpnnen 16275 resdomq 16292 aleph1re 16293 aleph1irr 16294 1nprm 16726 ensucne0OLD 43492 |
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