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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 9100 | . 2 ⊢ ω ∈ V | |
2 | isfiniteg 8772 | . 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 208 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 ωcom 7574 ≺ csdm 8502 Fincfn 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 |
This theorem is referenced by: fict 9110 infxpenlem 9433 pwsdompw 9620 cflim2 9679 axcc4dom 9857 domtriom 9859 fin41 9860 dominf 9861 infinf 9982 dominfac 9989 canthp1lem2 10069 pwfseqlem3 10076 pwfseqlem4a 10077 pwfseqlem4 10078 gchpwdom 10086 gchaleph 10087 gchhar 10095 omina 10107 gchina 10115 tskpr 10186 rexpen 15575 odinf 18684 fctop2 21607 dis1stc 22101 iunmbl2 24152 dyadmbl 24195 f1ocnt 30519 sibfof 31593 pibt2 34692 mblfinlem1 34923 ovoliunnfl 34928 heiborlem3 35085 ctbnfien 39408 pellex 39425 numinfctb 39696 saluncl 42596 meadjun 42738 meaiunlelem 42744 omeunle 42792 |
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