Theorem 2pwuninelg 6262

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 4538	. 2 ⊢ ¬ (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ∧ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
2		pwuni 4178	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		elpwg 3574	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
4	2, 3	mpbiri 167	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝐴)
5		ax-ia3 107	. . 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 659	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∈ wcel 2141   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797

This theorem is referenced by:  mnfnre  7962