Theorem 2pwuninelg 5839

Description: The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by Jim Kingdon, 14-Jan-2020.)

Assertion

Ref	Expression
2pwuninelg	⊢ (A ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ A ∈ A)

Proof of Theorem 2pwuninelg

Step	Hyp	Ref	Expression
1		en2lp 4232	. 2 ⊢ ¬ (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A)
2		pwuni 3934	. . . 4 ⊢ A ⊆ 𝒫 ∪ A
3		elpwg 3359	. . . 4 ⊢ (A ∈ 𝑉 → (A ∈ 𝒫 𝒫 ∪ A ↔ A ⊆ 𝒫 ∪ A))
4	2, 3	mpbiri 157	. . 3 ⊢ (A ∈ 𝑉 → A ∈ 𝒫 𝒫 ∪ A)
5		ax-ia3 101	. . 3 ⊢ (A ∈ 𝒫 𝒫 ∪ A → (𝒫 𝒫 ∪ A ∈ A → (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A)))
6	4, 5	syl 14	. 2 ⊢ (A ∈ 𝑉 → (𝒫 𝒫 ∪ A ∈ A → (A ∈ 𝒫 𝒫 ∪ A ∧ 𝒫 𝒫 ∪ A ∈ A)))
7	1, 6	mtoi 589	1 ⊢ (A ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ A ∈ A)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351  ∪ cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572

This theorem is referenced by:  mnfnre  6865