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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 5439 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴) | |
2 | snfi 9067 | . . . 4 ⊢ {𝑥} ∈ Fin | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ Fin) |
4 | 1, 3 | elind 4188 | . 2 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ (𝒫 𝐴 ∩ Fin)) |
5 | sneqbg 4840 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) | |
6 | 5 | adantr 479 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ({𝑥} = {𝑦} ↔ 𝑥 = 𝑦)) |
7 | 4, 6 | dom2 9014 | 1 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3938 𝒫 cpw 4598 {csn 4624 class class class wbr 5143 ≼ cdom 8960 Fincfn 8962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7869 df-1o 8485 df-en 8963 df-dom 8964 df-fin 8966 |
This theorem is referenced by: infpwfien 10085 ttukeylem1 10532 canthnum 10672 |
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