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Mirrors > Home > MPE Home > Th. List > isfin32i | Structured version Visualization version GIF version |
Description: One half of isfin3-2 9791. (Contributed by Mario Carneiro, 3-Jun-2015.) |
Ref | Expression |
---|---|
isfin32i | ⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin3 9720 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | |
2 | isfin4-2 9738 | . . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴)) | |
3 | 2 | ibi 269 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝒫 𝐴) |
4 | relwdom 9032 | . . . . . 6 ⊢ Rel ≼* | |
5 | 4 | brrelex1i 5610 | . . . . 5 ⊢ (ω ≼* 𝐴 → ω ∈ V) |
6 | canth2g 8673 | . . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω) | |
7 | sdomdom 8539 | . . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 ⊢ (ω ≼* 𝐴 → ω ≼ 𝒫 ω) |
9 | wdompwdom 9044 | . . . 4 ⊢ (ω ≼* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴) | |
10 | domtr 8564 | . . . 4 ⊢ ((ω ≼ 𝒫 ω ∧ 𝒫 ω ≼ 𝒫 𝐴) → ω ≼ 𝒫 𝐴) | |
11 | 8, 9, 10 | syl2anc 586 | . . 3 ⊢ (ω ≼* 𝐴 → ω ≼ 𝒫 𝐴) |
12 | 3, 11 | nsyl 142 | . 2 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼* 𝐴) |
13 | 1, 12 | sylbi 219 | 1 ⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 Vcvv 3496 𝒫 cpw 4541 class class class wbr 5068 ωcom 7582 ≼ cdom 8509 ≺ csdm 8510 ≼* cwdom 9023 FinIVcfin4 9704 FinIIIcfin3 9705 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-wdom 9025 df-fin4 9711 df-fin3 9712 |
This theorem is referenced by: isf33lem 9790 isfin3-2 9791 fin33i 9793 |
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