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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 10320 | . . 3 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | |
2 | isfin4-2 10338 | . . . . 5 ⊢ (𝒫 𝐴 ∈ FinIV → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴)) | |
3 | 2 | ibi 267 | . . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝒫 𝐴) |
4 | reldom 8970 | . . . . . . . 8 ⊢ Rel ≼ | |
5 | 4 | brrelex2i 5735 | . . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V) |
6 | canth2g 9156 | . . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≺ 𝒫 𝐴) |
8 | domsdomtr 9137 | . . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → ω ≺ 𝒫 𝐴) | |
9 | 7, 8 | mpdan 686 | . . . . 5 ⊢ (ω ≼ 𝐴 → ω ≺ 𝒫 𝐴) |
10 | sdomdom 9001 | . . . . 5 ⊢ (ω ≺ 𝒫 𝐴 → ω ≼ 𝒫 𝐴) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (ω ≼ 𝐴 → ω ≼ 𝒫 𝐴) |
12 | 3, 11 | nsyl 140 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝐴) |
13 | 1, 12 | sylbi 216 | . 2 ⊢ (𝐴 ∈ FinIII → ¬ ω ≼ 𝐴) |
14 | isfin4-2 10338 | . 2 ⊢ (𝐴 ∈ FinIII → (𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝐴)) | |
15 | 13, 14 | mpbird 257 | 1 ⊢ (𝐴 ∈ FinIII → 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2099 Vcvv 3471 𝒫 cpw 4603 class class class wbr 5148 ωcom 7870 ≼ cdom 8962 ≺ csdm 8963 FinIVcfin4 10304 FinIIIcfin3 10305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin4 10311 df-fin3 10312 |
This theorem is referenced by: finngch 10679 fin2so 37080 |
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