Theorem pwuninel2 6061

Description: 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 Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
pwuninel2	⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)

Proof of Theorem pwuninel2

Step	Hyp	Ref	Expression
1		pwnss 4000	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
2		elssuni 3687	. 2 ⊢ (𝒫 ∪ 𝐴 ∈ 𝐴 → 𝒫 ∪ 𝐴 ⊆ ∪ 𝐴)
3	1, 2	nsyl 594	1 ⊢ (∪ 𝐴 ∈ 𝑉 → ¬ 𝒫 ∪ 𝐴 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1439   ⊆ wss 3000  𝒫 cpw 3433  ∪ cuni 3659

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-nel 2352  df-rab 2369  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435  df-uni 3660

This theorem is referenced by:  pnfnre  7583