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Mirrors > Home > MPE Home > Th. List > Mathboxes > pwinfi | Structured version Visualization version GIF version |
Description: The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.) |
Ref | Expression |
---|---|
pwinfi | ⊢ (𝐴 ∈ (V ∖ Fin) ↔ 𝒫 𝐴 ∈ (V ∖ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vuniex 7744 | . . . 4 ⊢ ∪ 𝑥 ∈ V | |
2 | vpwex 5377 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
3 | 1, 2 | pm3.2i 470 | . . 3 ⊢ (∪ 𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) |
4 | 3 | rgenw 3062 | . 2 ⊢ ∀𝑥 ∈ V (∪ 𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) |
5 | pwinfig 42991 | . 2 ⊢ (∀𝑥 ∈ V (∪ 𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) → (𝐴 ∈ (V ∖ Fin) ↔ 𝒫 𝐴 ∈ (V ∖ Fin))) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ (𝐴 ∈ (V ∖ Fin) ↔ 𝒫 𝐴 ∈ (V ∖ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∀wral 3058 Vcvv 3471 ∖ cdif 3944 𝒫 cpw 4603 ∪ cuni 4908 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-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8487 df-en 8965 df-fin 8968 |
This theorem is referenced by: (None) |
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