Theorem pwinfig 43988

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 43987	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
2		pwfi 9229	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
3	2	notbii 320	. . . 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 633	. 2 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ Fin) ↔ (𝒫 𝐴 ∈ 𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin)))
6		eldif 3899	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ Fin))
7		eldif 3899	. 2 ⊢ (𝒫 𝐴 ∈ (𝐵 ∖ Fin) ↔ (𝒫 𝐴 ∈ 𝐵 ∧ ¬ 𝒫 𝐴 ∈ Fin))
8	5, 6, 7	3bitr4g 314	1 ⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3051   ∖ cdif 3886  𝒫 cpw 4541  ∪ cuni 4850  Fincfn 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-en 8894  df-dom 8895  df-fin 8897

This theorem is referenced by:  pwinfi2  43989  pwinfi3  43990  pwinfi  43991