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Mirrors > Home > MPE Home > Th. List > infpwfidom | Structured version Visualization version GIF version |
Description: The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
infpwfidom | ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snelpwi 5144 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
2 | snfi 8326 | . . . 4 ⊢ {𝑥} ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ Fin) |
4 | 1, 3 | elind 4020 | . 2 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ (𝒫 𝐴 ∩ Fin)) |
5 | sneqbg 4603 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) | |
6 | 5 | adantr 474 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
7 | 4, 6 | dom2 8284 | 1 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∩ cin 3790 𝒫 cpw 4378 {csn 4397 class class class wbr 4886 ≼ cdom 8239 Fincfn 8241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-1o 7843 df-en 8242 df-dom 8243 df-fin 8245 |
This theorem is referenced by: infpwfien 9218 ttukeylem1 9666 canthnum 9806 |
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