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Mirrors > Home > MPE Home > Th. List > fin34 | Structured version Visualization version GIF version |
Description: Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
Ref | Expression |
---|---|
fin34 | ⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin3 9404 | . . 3 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | |
2 | isfin4-2 9422 | . . . . 5 ⊢ (𝒫 𝐴 ∈ FinIV → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴)) | |
3 | 2 | ibi 259 | . . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝒫 𝐴) |
4 | reldom 8199 | . . . . . . . 8 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5362 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | canth2g 8354 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≺ 𝒫 𝐴) |
8 | domsdomtr 8335 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → ω ≺ 𝒫 𝐴) | |
9 | 7, 8 | mpdan 679 | . . . . 5 ⊢ (ω ≼ 𝐴 → ω ≺ 𝒫 𝐴) |
10 | sdomdom 8221 | . . . . 5 ⊢ (ω ≺ 𝒫 𝐴 → ω ≼ 𝒫 𝐴) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (ω ≼ 𝐴 → ω ≼ 𝒫 𝐴) |
12 | 3, 11 | nsyl 138 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝐴) |
13 | 1, 12 | sylbi 209 | . 2 ⊢ (𝐴 ∈ FinIII → ¬ ω ≼ 𝐴) |
14 | isfin4-2 9422 | . 2 ⊢ (𝐴 ∈ FinIII → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴)) | |
15 | 13, 14 | mpbird 249 | 1 ⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2157 Vcvv 3383 𝒫 cpw 4347 class class class wbr 4841 ωcom 7297 ≼ cdom 8191 ≺ csdm 8192 FinIVcfin4 9388 FinIIIcfin3 9389 |
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 2375 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-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-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-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-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fin4 9395 df-fin3 9396 |
This theorem is referenced by: finngch 9763 fin2so 33876 |
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