Theorem pwuniss 29370

Description: Condition for a class union to be a subset. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Assertion

Ref	Expression
pwuniss	⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)

Proof of Theorem pwuniss

Step	Hyp	Ref	Expression
1		uniss 4458	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ ∪ 𝒫 𝐵)
2		unipw 4918	. 2 ⊢ ∪ 𝒫 𝐵 = 𝐵
3	1, 2	syl6sseq 3651	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 → ∪ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  elpwunicl  29371  pwldsys  30220