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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 40043 | . 2 ⊢ ∅ ∈ (𝒫 𝐴 ∩ Fin) | |
2 | ne0i 4149 | . 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 2166 ≠ wne 2998 ∩ cin 3796 ∅c0 4143 𝒫 cpw 4377 Fincfn 8221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-om 7326 df-en 8222 df-fin 8225 |
This theorem is referenced by: sge0z 41382 |
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