Theorem pwuni 4911

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 4904	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		velpw 4565	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
3	1, 2	sylibr 237	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
4	3	ssriv 3917	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  ∪ cuni 4872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-pw 4562  df-uni 4873

This theorem is used by:  uniexr  7809  fipwuni  9455  uniwf  9865  rankuni  9916  rankc2  9925  rankxplim  9933  fin23lem17  10455  axcclem  10574  grurn  10925  istopon  23285  eltg3i  23334  cmpfi  23781  hmphdis  24170  ptcmpfi  24187  fbssfi  24211  mopnfss  24817  pliguhgr  31145  shsspwh  31905  circtopn  34580  hasheuni  34828  issgon  34866  sigaclci  34875  sigagenval  34884  dmsigagen  34888  imambfm  35005  bj-unirel  38236  salgenval  47604  salgenn0  47614  caragensspw  47792