Theorem pwuni 4943

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 4935	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		velpw 4603	. . 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 3944  𝒫 cpw 4598  ∪ cuni 4903

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 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-in 3951  df-ss 3961  df-pw 4600  df-uni 4904

This theorem is referenced by:  uniexr  7759  fipwuni  9441  uniwf  9834  rankuni  9878  rankc2  9886  rankxplim  9894  fin23lem17  10353  axcclem  10472  grurn  10816  istopon  22801  eltg3i  22851  cmpfi  23299  hmphdis  23687  ptcmpfi  23704  fbssfi  23728  mopnfss  24336  pliguhgr  30283  shsspwh  31043  circtopn  33374  hasheuni  33640  issgon  33678  sigaclci  33687  sigagenval  33695  dmsigagen  33699  imambfm  33818  bj-unirel  36466  salgenval  45632  salgenn0  45642  caragensspw  45820