Theorem pwuni 4945

Description: A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)

Assertion

Ref	Expression
pwuni	⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Proof of Theorem pwuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elssuni 4937	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		velpw 4602	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
3	1, 2	sylibr 233	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
4	3	ssriv 3982	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2099   ⊆ wss 3946  𝒫 cpw 4597  ∪ cuni 4905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-ss 3963  df-pw 4599  df-uni 4906

This theorem is referenced by:  uniexr  7763  fipwuni  9462  uniwf  9855  rankuni  9899  rankc2  9907  rankxplim  9915  fin23lem17  10372  axcclem  10491  grurn  10835  istopon  22902  eltg3i  22952  cmpfi  23400  hmphdis  23788  ptcmpfi  23805  fbssfi  23829  mopnfss  24437  pliguhgr  30416  shsspwh  31176  circtopn  33665  hasheuni  33931  issgon  33969  sigaclci  33978  sigagenval  33986  dmsigagen  33990  imambfm  34109  bj-unirel  36771  salgenval  45978  salgenn0  45988  caragensspw  46166