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Theorem pwinfig 38392
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 38391 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
2 pwfi 8417 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32notbii 309 . . . 4 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin)
43a1i 11 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (¬ 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin))
51, 4anbi12d 616 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ((𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin)))
6 eldif 3733 . 2 (𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin))
7 eldif 3733 . 2 (𝒫 𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin))
85, 6, 73bitr4g 303 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 196  wa 382  wcel 2145  wral 3061  cdif 3720  𝒫 cpw 4297   cuni 4574  Fincfn 8109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113
This theorem is referenced by:  pwinfi2  38393  pwinfi3  38394  pwinfi  38395
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