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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 9108 | . 2 ⊢ ω ∈ V | |
2 | isfiniteg 8780 | . 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 2114 Vcvv 3496 class class class wbr 5068 ωcom 7582 ≺ csdm 8510 Fincfn 8511 |
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 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 |
This theorem is referenced by: fict 9118 infxpenlem 9441 pwsdompw 9628 cflim2 9687 axcc4dom 9865 domtriom 9867 fin41 9868 dominf 9869 infinf 9990 dominfac 9997 canthp1lem2 10077 pwfseqlem3 10084 pwfseqlem4a 10085 pwfseqlem4 10086 gchpwdom 10094 gchaleph 10095 gchhar 10103 omina 10115 gchina 10123 tskpr 10194 rexpen 15583 odinf 18692 fctop2 21615 dis1stc 22109 iunmbl2 24160 dyadmbl 24203 f1ocnt 30527 sibfof 31600 pibt2 34700 mblfinlem1 34931 ovoliunnfl 34936 heiborlem3 35093 ctbnfien 39422 pellex 39439 numinfctb 39710 saluncl 42609 meadjun 42751 meaiunlelem 42757 omeunle 42805 |
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