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Mirrors > Home > MPE Home > Th. List > isfinite | Structured version Visualization version GIF version |
Description: A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.) |
Ref | Expression |
---|---|
isfinite | ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 9181 | . 2 ⊢ ω ∈ V | |
2 | isfiniteg 8854 | . 2 ⊢ (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2114 Vcvv 3398 class class class wbr 5030 ωcom 7601 ≺ csdm 8556 Fincfn 8557 |
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 7481 ax-inf2 9179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 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-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 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-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7602 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 |
This theorem is referenced by: fict 9191 infxpenlem 9515 pwsdompw 9706 cflim2 9765 axcc4dom 9943 domtriom 9945 fin41 9946 dominf 9947 infinf 10068 dominfac 10075 canthp1lem2 10155 pwfseqlem3 10162 pwfseqlem4a 10163 pwfseqlem4 10164 gchpwdom 10172 gchaleph 10173 gchhar 10181 omina 10193 gchina 10201 tskpr 10272 rexpen 15675 odinf 18810 fctop2 21758 dis1stc 22252 iunmbl2 24311 dyadmbl 24354 f1ocnt 30700 sibfof 31879 pibt2 35233 mblfinlem1 35459 ovoliunnfl 35464 heiborlem3 35616 ctbnfien 40234 pellex 40251 numinfctb 40522 saluncl 43422 meadjun 43564 meaiunlelem 43570 omeunle 43618 |
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