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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 44896 | . 2 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) | |
2 | ne0i 4359 | . 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 2103 ≠ wne 2942 ∩ cin 3969 ∅c0 4347 𝒫 cpw 4622 Fincfn 8999 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-ord 6397 df-on 6398 df-lim 6399 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-om 7900 df-en 9000 df-fin 9003 |
This theorem is referenced by: sge0z 46231 |
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