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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 8993 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
2 | nnsdomg 9323 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → 𝑥 ≺ ω) | |
3 | sdomen1 9142 | . . . . 5 ⊢ (𝐴 ≈ 𝑥 → (𝐴 ≺ ω ↔ 𝑥 ≺ ω)) | |
4 | 2, 3 | syl5ibrcom 246 | . . . 4 ⊢ ((ω ∈ V ∧ 𝑥 ∈ ω) → (𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
5 | 4 | rexlimdva 3145 | . . 3 ⊢ (ω ∈ V → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 → 𝐴 ≺ ω)) |
6 | 1, 5 | biimtrid 241 | . 2 ⊢ (ω ∈ V → (𝐴 ∈ Fin → 𝐴 ≺ ω)) |
7 | isfinite2 9322 | . 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 394 ∈ wcel 2098 ∃wrex 3060 Vcvv 3463 class class class wbr 5141 ωcom 7866 ≈ cen 8957 ≺ csdm 8959 Fincfn 8960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-om 7867 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 |
This theorem is referenced by: unfi2 9337 unifi2 9364 isfinite 9673 axcclem 10478 |
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