isfin32i - Metamath Proof Explorer

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Theorem isfin32i 10370

Description: One half of isfin3-2 10372. (Contributed by Mario Carneiro, 3-Jun-2015.)

**Assertion**
Ref	Expression
isfin32i	⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)

**Proof of Theorem isfin32i**
Step	Hyp	Ref	Expression
1		isfin3 10301	. 2 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10319	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 270	. . 3 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝒫 𝐴)
4		relwdom 9541	. . . . . 6 ⊢ Rel ≼^*
5	4	brrelex1i 5711	. . . . 5 ⊢ (ω ≼^* 𝐴 → ω ∈ V)
6		canth2g 9132	. . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω)
7		sdomdom 8989	. . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω)
8	5, 6, 7	3syl 19	. . . 4 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 ω)
9		wdompwdom 9553	. . . 4 ⊢ (ω ≼^* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴)
10		domtr 9016	. . . 4 ⊢ ((ω ≼ 𝒫 ω ∧ 𝒫 ω ≼ 𝒫 𝐴) → ω ≼ 𝒫 𝐴)
11	8, 9, 10	syl2anc 596	. . 3 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 𝐴)
12	3, 11	nsyl 141	. 2 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼^* 𝐴)
13	1, 12	sylbi 220	1 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∈ wcel 2145 Vcvv 3450 𝒫 cpw 4557 class class class wbr 5103 ωcom 7863 ≼ cdom 8953 ≺ csdm 8954 ≼^* cwdom 9539 Fin^IVcfin4 10285 Fin^IIIcfin3 10286

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-wdom 9540 df-fin4 10292 df-fin3 10293

This theorem is used by: isf33lem 10371 isfin3-2 10372 fin33i 10374

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