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Theorem fin34 9658

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 9564	. . 3 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 9582	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 268	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝒫 𝐴)
4		reldom 8363	. . . . . . . 8 ⊢ Rel ≼
5	4	brrelex2i 5495	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
6		canth2g 8518	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
7	5, 6	syl 17	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≺ 𝒫 𝐴)
8		domsdomtr 8499	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → ω ≺ 𝒫 𝐴)
9	7, 8	mpdan 683	. . . . 5 ⊢ (ω ≼ 𝐴 → ω ≺ 𝒫 𝐴)
10		sdomdom 8385	. . . . 5 ⊢ (ω ≺ 𝒫 𝐴 → ω ≼ 𝒫 𝐴)
11	9, 10	syl 17	. . . 4 ⊢ (ω ≼ 𝐴 → ω ≼ 𝒫 𝐴)
12	3, 11	nsyl 142	. . 3 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝐴)
13	1, 12	sylbi 218	. 2 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼ 𝐴)
14		isfin4-2 9582	. 2 ⊢ (𝐴 ∈ Fin^III → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
15	13, 14	mpbird 258	1 ⊢ (𝐴 ∈ Fin^III → 𝐴 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2081 Vcvv 3437 𝒫 cpw 4453 class class class wbr 4962 ωcom 7436 ≼ cdom 8355 ≺ csdm 8356 Fin^IVcfin4 9548 Fin^IIIcfin3 9549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fin4 9555 df-fin3 9556

This theorem is referenced by: finngch 9923 fin2so 34429

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