fin34 - Metamath Proof Explorer

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Theorem fin34 10389

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 10295	. . 3 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10313	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 270	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝒫 𝐴)
4		reldom 8955	. . . . . . . 8 ⊢ Rel ≼
5	4	brrelex2i 5720	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
6		canth2g 9126	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
7	5, 6	syl 18	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ≺ 𝒫 𝐴)
8		domsdomtr 9107	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ≺ 𝒫 𝐴) → ω ≺ 𝒫 𝐴)
9	7, 8	mpdan 700	. . . . 5 ⊢ (ω ≼ 𝐴 → ω ≺ 𝒫 𝐴)
10		sdomdom 8983	. . . . 5 ⊢ (ω ≺ 𝒫 𝐴 → ω ≼ 𝒫 𝐴)
11	9, 10	syl 18	. . . 4 ⊢ (ω ≼ 𝐴 → ω ≼ 𝒫 𝐴)
12	3, 11	nsyl 141	. . 3 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝐴)
13	1, 12	sylbi 220	. 2 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼ 𝐴)
14		isfin4-2 10313	. 2 ⊢ (𝐴 ∈ Fin^III → (𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝐴))
15	13, 14	mpbird 260	1 ⊢ (𝐴 ∈ Fin^III → 𝐴 ∈ Fin^IV)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∈ wcel 2146 Vcvv 3457 𝒫 cpw 4564 class class class wbr 5111 ωcom 7868 ≼ cdom 8947 ≺ csdm 8948 Fin^IVcfin4 10279 Fin^IIIcfin3 10280

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin4 10286 df-fin3 10287

This theorem is used by: finngch 10657 fin2so 38317

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