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Mirrors > Home > MPE Home > Th. List > fin11a | Structured version Visualization version GIF version |
Description: Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
fin11a | ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinIa) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4506 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | ssfi 8722 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
3 | 1, 2 | sylan2 595 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ Fin) |
4 | 3 | orcd 870 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝒫 𝐴) → (𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)) |
5 | 4 | ralrimiva 3149 | . 2 ⊢ (𝐴 ∈ Fin → ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin)) |
6 | isfin1a 9703 | . 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑥 ∈ Fin ∨ (𝐴 ∖ 𝑥) ∈ Fin))) | |
7 | 5, 6 | mpbird 260 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinIa) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ⊆ wss 3881 𝒫 cpw 4497 Fincfn 8492 FinIacfin1a 9689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-om 7561 df-er 8272 df-en 8493 df-fin 8496 df-fin1a 9696 |
This theorem is referenced by: (None) |
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