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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwfin0 | Structured version Visualization version GIF version |
Description: A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
pwfin0 | ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0pwfi 44695 | . 2 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) | |
2 | ne0i 4335 | . 2 ⊢ (∅ ∈ (𝒫 𝐴 ∩ Fin) → (𝒫 𝐴 ∩ Fin) ≠ ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝒫 𝐴 ∩ Fin) ≠ ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ≠ wne 2930 ∩ cin 3946 ∅c0 4323 𝒫 cpw 4598 Fincfn 8964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-ord 6369 df-on 6370 df-lim 6371 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-om 7867 df-en 8965 df-fin 8968 |
This theorem is referenced by: sge0z 46030 |
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