Theorem 2pwuninelg 6302

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 4568	. 2 ⊢ ¬ (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ∧ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)
2		pwuni 4207	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
3		elpwg 3598	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 ∪ 𝐴 ↔ 𝐴 ⊆ 𝒫 ∪ 𝐴))
4	2, 3	mpbiri 168	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝒫 ∪ 𝐴)
5		ax-ia3 108	. . 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 665	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2160   ⊆ wss 3144  𝒫 cpw 3590  ∪ cuni 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825

This theorem is referenced by:  mnfnre  8019