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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0pwfi | Structured version Visualization version GIF version |
Description: The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
0pwfi | ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elpw 5344 | . 2 ⊢ ∅ ∈ 𝒫 𝐴 | |
2 | 0fin 9167 | . 2 ⊢ ∅ ∈ Fin | |
3 | 1, 2 | elini 4185 | 1 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ∩ cin 3939 ∅c0 4314 𝒫 cpw 4594 Fincfn 8935 |
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-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-om 7849 df-en 8936 df-fin 8939 |
This theorem is referenced by: pwfin0 44237 |
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