isfin32i - Metamath Proof Explorer

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Theorem isfin32i 10277

Description: One half of isfin3-2 10279. (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 10208	. 2 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10226	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin^IV → (𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝐴))
3	2	ibi 267	. . 3 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼ 𝒫 𝐴)
4		relwdom 9473	. . . . . 6 ⊢ Rel ≼^*
5	4	brrelex1i 5680	. . . . 5 ⊢ (ω ≼^* 𝐴 → ω ∈ V)
6		canth2g 9061	. . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω)
7		sdomdom 8919	. . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω)
8	5, 6, 7	3syl 18	. . . 4 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 ω)
9		wdompwdom 9485	. . . 4 ⊢ (ω ≼^* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴)
10		domtr 8946	. . . 4 ⊢ ((ω ≼ 𝒫 ω ∧ 𝒫 ω ≼ 𝒫 𝐴) → ω ≼ 𝒫 𝐴)
11	8, 9, 10	syl2anc 584	. . 3 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 𝐴)
12	3, 11	nsyl 140	. 2 ⊢ (𝒫 𝐴 ∈ Fin^IV → ¬ ω ≼^* 𝐴)
13	1, 12	sylbi 217	1 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2113 Vcvv 3440 𝒫 cpw 4554 class class class wbr 5098 ωcom 7808 ≼ cdom 8883 ≺ csdm 8884 ≼^* cwdom 9471 Fin^IVcfin4 10192 Fin^IIIcfin3 10193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-wdom 9472 df-fin4 10199 df-fin3 10200

This theorem is referenced by: isf33lem 10278 isfin3-2 10279 fin33i 10281

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