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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 8021 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnsdomg 8260 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω) | |
3 | sdomen1 8145 | . . . . 5 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω)) | |
4 | 2, 3 | syl5ibrcom 237 | . . . 4 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
5 | 4 | rexlimdva 3060 | . . 3 ⊢ (ω ∈ V → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
6 | 1, 5 | syl5bi 232 | . 2 ⊢ (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω)) |
7 | isfinite2 8259 | . 2 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
8 | 6, 7 | impbid1 215 | 1 ⊢ (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2030 ∃wrex 2942 Vcvv 3231 class class class wbr 4685 ωcom 7107 ≈ cen 7994 ≺ csdm 7996 Fincfn 7997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 |
This theorem is referenced by: unfi2 8270 unifi2 8297 isfinite 8587 axcclem 9317 |
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