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Theorem fin34 10414

Description: Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)

**Assertion**
Ref	Expression
fin34	⊢ (𝐴 ∈ Fin^III → 𝐴 ∈ Fin^IV)

**Proof of Theorem fin34**
Step	Hyp	Ref	Expression
1		isfin3 10320	. . 3 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10338	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 267	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ 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 ∈ Fin^IV → ¬ ω ≼ 𝐴)
13	1, 12	sylbi 216	. 2 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼ 𝐴)
14		isfin4-2 10338	. 2 ⊢ (𝐴 ∈ Fin^III → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
15	13, 14	mpbird 257	1 ⊢ (𝐴 ∈ Fin^III → 𝐴 ∈ Fin^IV)

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 Fin^IVcfin4 10304 Fin^IIIcfin3 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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