Theorem pwinfig 44012

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	⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwinfig

Step	Hyp	Ref	Expression
1		pwelg 44011	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
2		pwfi 9226	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	notbii 321	. . . 4 ⊢ (¬ 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin)
4	3	a1i 11	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (¬ 𝐴 ∈ Fin ↔ ¬ 𝒫 𝐴 ∈ Fin))
5	1, 4	anbi12d 638	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ Fin) ↔ (𝒫 𝐴 ∈ 𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin)))
6		eldif 3900	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ Fin))
7		eldif 3900	. 2 ⊢ (𝒫 𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝒫 𝐴 ∈ 𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin))
8	5, 6, 7	3bitr4g 315	1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∀wral 3054   ∖ cdif 3887  𝒫 cpw 4536  ∪ cuni 4845  Fincfn 8890

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7814  df-1o 8402  df-en 8891  df-dom 8892  df-fin 8894

This theorem is referenced by:  pwinfi2  44013  pwinfi3  44014  pwinfi  44015