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| Mirrors > Home > MPE Home > Th. List > endom | Structured version Visualization version GIF version | ||
| Description: Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.) |
| Ref | Expression |
|---|---|
| endom | ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | enssdom 9017 | . 2 ⊢ ≈ ⊆ ≼ | |
| 2 | 1 | ssbri 5188 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: bren2 9023 domrefg 9027 endomtr 9052 domentr 9053 domunsncan 9112 sbthb 9134 dom0 9142 sdomentr 9151 ensdomtr 9153 domtriord 9163 domunsn 9167 xpen 9180 sdomdomtrfi 9241 domsdomtrfi 9242 sucdom2 9243 php 9247 php3 9249 onomeneq 9265 0sdom1dom 9274 rex2dom 9282 unxpdom2 9290 sucxpdom 9291 f1finf1o 9305 findcard3 9318 fodomfi 9350 wdomen1 9616 wdomen2 9617 fidomtri2 10034 prdom2 10046 acnen 10093 acnen2 10095 alephdom 10121 alephinit 10135 undjudom 10208 pwdjudom 10255 fin1a2lem11 10450 hsmexlem1 10466 gchdomtri 10669 gchdjuidm 10708 gchxpidm 10709 gchpwdom 10710 gchhar 10719 gruina 10858 nnct 14022 odinf 19581 hauspwdom 23509 ufildom1 23934 iscmet3 25327 mbfaddlem 25695 ctbssinf 37407 pibt2 37418 heiborlem3 37820 zct 45066 qct 45373 caratheodory 46543 |
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