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Mirrors > Home > MPE Home > Th. List > isfin32i | Structured version Visualization version GIF version |
Description: One half of isfin3-2 9788. (Contributed by Mario Carneiro, 3-Jun-2015.) |
Ref | Expression |
---|---|
isfin32i | ⊢ (𝐴 ∈ FinIII → ¬ ω ≼* 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin3 9717 | . 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV) | |
2 | isfin4-2 9735 | . . . 4 ⊢ (𝒫 𝐴 ∈ FinIV → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴)) | |
3 | 2 | ibi 269 | . . 3 ⊢ (𝒫 𝐴 ∈ FinIV → ¬ ω ≼ 𝒫 𝐴) |
4 | relwdom 9029 | . . . . . 6 ⊢ Rel ≼* | |
5 | 4 | brrelex1i 5607 | . . . . 5 ⊢ (ω ≼* 𝐴 → ω ∈ V) |
6 | canth2g 8670 | . . . . 5 ⊢ (ω ∈ V → ω ≺ 𝒫 ω) | |
7 | sdomdom 8536 | . . . . 5 ⊢ (ω ≺ 𝒫 ω → ω ≼ 𝒫 ω) | |
8 | 5, 6, 7 | 3syl 18 | . . . 4 ⊢ (ω ≼* 𝐴 → ω ≼ 𝒫 ω) |
9 | wdompwdom 9041 | . . . 4 ⊢ (ω ≼* 𝐴 → 𝒫 ω ≼ 𝒫 𝐴) | |
10 | domtr 8561 | . . . 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 2110 Vcvv 3494 𝒫 cpw 4538 class class class wbr 5065 ωcom 7579 ≼ cdom 8506 ≺ csdm 8507 ≼* cwdom 9020 FinIVcfin4 9701 FinIIIcfin3 9702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-wdom 9022 df-fin4 9708 df-fin3 9709 |
This theorem is referenced by: isf33lem 9787 isfin3-2 9788 fin33i 9790 |
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