isfin32i - Metamath Proof Explorer

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Theorem isfin32i 10121

Description: One half of isfin3-2 10123. (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 10052	. 2 ⊢ (𝐴 ∈ Fin^III ↔ 𝒫 𝐴 ∈ Fin^IV)
2		isfin4-2 10070	. . . 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 9325	. . . . . 6 ⊢ Rel ≼^*
5	4	brrelex1i 5643	. . . . 5 ⊢ (ω ≼^* 𝐴 → ω ∈ V)
6		canth2g 8918	. . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω)
7		sdomdom 8768	. . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω)
8	5, 6, 7	3syl 18	. . . 4 ⊢ (ω ≼^* 𝐴 → ω ≼ 𝒫 ω)
9		wdompwdom 9337	. . . 4 ⊢ (ω ≼^* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴)
10		domtr 8793	. . . 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 216	1 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 Vcvv 3432 𝒫 cpw 4533 class class class wbr 5074 ωcom 7712 ≼ cdom 8731 ≺ csdm 8732 ≼^* cwdom 9323 Fin^IVcfin4 10036 Fin^IIIcfin3 10037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-wdom 9324 df-fin4 10043 df-fin3 10044

This theorem is referenced by: isf33lem 10122 isfin3-2 10123 fin33i 10125

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