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Theorem fin34 10319

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 10225	. . 3 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10243	. . . . 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 8901	. . . . . . . 8 ⊢ Rel ≼
5	4	brrelex2i 5688	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
6		canth2g 9072	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
7	5, 6	syl 17	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≺ 𝒫 𝐴)
8		domsdomtr 9053	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → ω ≺ 𝒫 𝐴)
9	7, 8	mpdan 687	. . . . 5 ⊢ (ω ≼ 𝐴 → ω ≺ 𝒫 𝐴)
10		sdomdom 8928	. . . . 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 217	. 2 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼ 𝐴)
14		isfin4-2 10243	. 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 2109 Vcvv 3444 𝒫 cpw 4559 class class class wbr 5102 ωcom 7822 ≼ cdom 8893 ≺ csdm 8894 Fin^IVcfin4 10209 Fin^IIIcfin3 10210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7372 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin4 10216 df-fin3 10217

This theorem is referenced by: finngch 10584 fin2so 37574

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