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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 8923 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnsdomg 9253 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω) | |
3 | sdomen1 9072 | . . . . 5 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω)) | |
4 | 2, 3 | syl5ibrcom 247 | . . . 4 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
5 | 4 | rexlimdva 3153 | . . 3 ⊢ (ω ∈ V → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
6 | 1, 5 | biimtrid 241 | . 2 ⊢ (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω)) |
7 | isfinite2 9252 | . 2 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
8 | 6, 7 | impbid1 224 | 1 ⊢ (ω ∈ V → (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∃wrex 3074 Vcvv 3448 class class class wbr 5110 ωcom 7807 ≈ cen 8887 ≺ csdm 8889 Fincfn 8890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 |
This theorem is referenced by: unfi2 9266 unifi2 9293 isfinite 9595 axcclem 10400 |
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