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Mirrors > Home > MPE Home > Th. List > fin1a2 | Structured version Visualization version GIF version |
Description: Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
fin1a2 | ⊢ (𝐴 ∈ FinIa → 𝐴 ∈ FinII) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4357 | . . . 4 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴) | |
2 | fin1ai 9401 | . . . . 5 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin)) | |
3 | fin12 9521 | . . . . . 6 ⊢ ((𝐴 ∖ 𝑏) ∈ Fin → (𝐴 ∖ 𝑏) ∈ FinII) | |
4 | 3 | orim2i 935 | . . . . 5 ⊢ ((𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ Fin) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ⊆ 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
6 | 1, 5 | sylan2 587 | . . 3 ⊢ ((𝐴 ∈ FinIa ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
7 | 6 | ralrimiva 3145 | . 2 ⊢ (𝐴 ∈ FinIa → ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) |
8 | fin1a2s 9522 | . 2 ⊢ ((𝐴 ∈ FinIa ∧ ∀𝑏 ∈ 𝒫 𝐴(𝑏 ∈ Fin ∨ (𝐴 ∖ 𝑏) ∈ FinII)) → 𝐴 ∈ FinII) | |
9 | 7, 8 | mpdan 679 | 1 ⊢ (𝐴 ∈ FinIa → 𝐴 ∈ FinII) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 ∈ wcel 2157 ∀wral 3087 ∖ cdif 3764 ⊆ wss 3767 𝒫 cpw 4347 Fincfn 8193 FinIacfin1a 9386 FinIIcfin2 9387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-rpss 7169 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-seqom 7780 df-1o 7797 df-2o 7798 df-oadd 7801 df-omul 7802 df-er 7980 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-wdom 8704 df-card 9049 df-fin1a 9393 df-fin2 9394 df-fin4 9395 df-fin3 9396 |
This theorem is referenced by: (None) |
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