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Theorem pwinfig 42994
Description: The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝐵 is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
Assertion
Ref Expression
pwinfig (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwinfig
StepHypRef Expression
1 pwelg 42993 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
2 pwfi 9207 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32notbii 319 . . . 4 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin)
43a1i 11 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (¬ 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin))
51, 4anbi12d 630 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ((𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin)))
6 eldif 3957 . 2 (𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin))
7 eldif 3957 . 2 (𝒫 𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin))
85, 6, 73bitr4g 313 1 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2098  wral 3057  cdif 3944  𝒫 cpw 4604   cuni 4910  Fincfn 8968
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-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-om 7875  df-1o 8491  df-en 8969  df-fin 8972
This theorem is referenced by:  pwinfi2  42995  pwinfi3  42996  pwinfi  42997
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