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Mirrors > Home > MPE Home > Th. List > isfiniteg | 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. In order to avoid the Axiom of infinity, we include it as a hypothesis. (Contributed by NM, 3-Nov-2002.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
isfiniteg | ⊢ (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 9014 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnsdomg 9332 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω) | |
3 | sdomen1 9159 | . . . . 5 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω)) | |
4 | 2, 3 | syl5ibrcom 247 | . . . 4 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
5 | 4 | rexlimdva 3152 | . . 3 ⊢ (ω ∈ V → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
6 | 1, 5 | biimtrid 242 | . 2 ⊢ (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω)) |
7 | isfinite2 9331 | . 2 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
8 | 6, 7 | impbid1 225 | 1 ⊢ (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 ∃wrex 3067 Vcvv 3477 class class class wbr 5147 ωcom 7886 ≈ cen 8980 ≺ csdm 8982 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 |
This theorem is referenced by: unfi2 9345 unifi2 9382 isfinite 9689 axcclem 10494 |
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