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Mirrors > Home > MPE Home > Th. List > isfin32i | Structured version Visualization version GIF version |
Description: One half of isfin3-2 9072. (Contributed by Mario Carneiro, 3-Jun-2015.) |
Ref | Expression |
---|---|
isfin32i | ⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin3 9001 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | |
2 | isfin4-2 9019 | . . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴)) | |
3 | 2 | ibi 255 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝒫 𝐴) |
4 | relwdom 8354 | . . . . . 6 ⊢ Rel ≼* | |
5 | 4 | brrelexi 5082 | . . . . 5 ⊢ (ω ≼* 𝐴 → ω ∈ V) |
6 | canth2g 7999 | . . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω) | |
7 | sdomdom 7869 | . . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 ⊢ (ω ≼* 𝐴 → ω ≼ 𝒫 ω) |
9 | wdompwdom 8366 | . . . 4 ⊢ (ω ≼* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴) | |
10 | domtr 7895 | . . . 4 ⊢ ((ω ≼ 𝒫 ω ∧ 𝒫 ω ≼ 𝒫 𝐴) → ω ≼ 𝒫 𝐴) | |
11 | 8, 9, 10 | syl2anc 691 | . . 3 ⊢ (ω ≼* 𝐴 → ω ≼ 𝒫 𝐴) |
12 | 3, 11 | nsyl 134 | . 2 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼* 𝐴) |
13 | 1, 12 | sylbi 206 | 1 ⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 ≺ csdm 7840 ≼* cwdom 8345 FinIVcfin4 8985 FinIIIcfin3 8986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-wdom 8347 df-fin4 8992 df-fin3 8993 |
This theorem is referenced by: isf33lem 9071 isfin3-2 9072 fin33i 9074 |
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