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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 8538 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
2 | domsdomtr 8654 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) | |
3 | 1, 2 | sylan 582 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 class class class wbr 5068 ≈ cen 8508 ≼ cdom 8509 ≺ csdm 8510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 |
This theorem is referenced by: sdomen1 8663 sucxpdom 8729 f1finf1o 8747 findcard3 8763 isfinite2 8778 pm54.43 9431 infxpenlem 9441 alephnbtwn2 9500 alephordi 9502 alephsucdom 9507 pwsdompw 9628 infunsdom1 9637 cflim2 9687 fin23lem27 9752 cfpwsdom 10008 inawinalem 10113 inar1 10199 tskcard 10205 tskuni 10207 rpnnen 15582 resdomq 15599 aleph1re 15600 aleph1irr 15601 1nprm 16025 ensucne0OLD 39903 |
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