isfin32i - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isfin32i		Structured version Visualization version GIF version

Theorem isfin32i 10362

Description: One half of isfin3-2 10364. (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 10293	. 2 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10311	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 266	. . 3 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝒫 𝐴)
4		relwdom 9563	. . . . . 6 ⊢ Rel ≼^*
5	4	brrelex1i 5731	. . . . 5 ⊢ (ω ≼^* 𝐴 → ω ∈ V)
6		canth2g 9133	. . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω)
7		sdomdom 8978	. . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω)
8	5, 6, 7	3syl 18	. . . 4 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 ω)
9		wdompwdom 9575	. . . 4 ⊢ (ω ≼^* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴)
10		domtr 9005	. . . 4 ⊢ ((ω ≼ 𝒫 ω ∧ 𝒫 ω ≼ 𝒫 𝐴) → ω ≼ 𝒫 𝐴)
11	8, 9, 10	syl2anc 582	. . 3 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 𝐴)
12	3, 11	nsyl 140	. 2 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼^* 𝐴)
13	1, 12	sylbi 216	1 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2104 Vcvv 3472 𝒫 cpw 4601 class class class wbr 5147 ωcom 7857 ≼ cdom 8939 ≺ csdm 8940 ≼^* cwdom 9561 Fin^IVcfin4 10277 Fin^IIIcfin3 10278

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-wdom 9562 df-fin4 10284 df-fin3 10285

This theorem is referenced by: isf33lem 10363 isfin3-2 10364 fin33i 10366

W3C validator