Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwinfig Structured version   Visualization version   GIF version

Theorem pwinfig 40713
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 40712 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (𝐴𝐵 ↔ 𝒫 𝐴𝐵))
2 pwfi 8776 . . . . 5 (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
32notbii 323 . . . 4 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin)
43a1i 11 . . 3 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → (¬ 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin))
51, 4anbi12d 634 . 2 (∀𝑥𝐵 ( 𝑥𝐵 ∧ 𝒫 𝑥𝐵) → ((𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin)))
6 eldif 3853 . 2 (𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝐴𝐵 ∧ ¬ 𝐴 ∈ Fin))
7 eldif 3853 . 2 (𝒫 𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝒫 𝐴𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin))
85, 6, 73bitr4g 317 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 209  wa 399  wcel 2114  wral 3053  cdif 3840  𝒫 cpw 4488   cuni 4796  Fincfn 8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-om 7600  df-1o 8131  df-en 8556  df-fin 8559
This theorem is referenced by:  pwinfi2  40714  pwinfi3  40715  pwinfi  40716
  Copyright terms: Public domain W3C validator