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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 9644 | . 2 ⊢ ω ∈ V | |
2 | isfiniteg 9310 | . 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 205 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 ωcom 7859 ≺ csdm 8944 Fincfn 8945 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 |
This theorem is referenced by: fict 9654 infxpenlem 10014 pwsdompw 10205 cflim2 10264 axcc4dom 10442 domtriom 10444 fin41 10445 dominf 10446 infinf 10567 dominfac 10574 canthp1lem2 10654 pwfseqlem3 10661 pwfseqlem4a 10662 pwfseqlem4 10663 gchpwdom 10671 gchaleph 10672 gchhar 10680 omina 10692 gchina 10700 tskpr 10771 rexpen 16178 odinf 19479 fctop2 22828 dis1stc 23323 iunmbl2 25406 dyadmbl 25449 f1ocnt 32446 sibfof 33803 pibt2 36762 mblfinlem1 36989 ovoliunnfl 36994 heiborlem3 37145 ctbnfien 42019 pellex 42036 numinfctb 42308 saluncl 45492 meadjun 45637 meaiunlelem 45643 omeunle 45691 |
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