Theorem pwuni 4913

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 4906	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		velpw 4569	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
3	1, 2	sylibr 237	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
4	3	ssriv 3942	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2146   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-pw 4566  df-uni 4875

This theorem is used by:  uniexr  7764  fipwuni  9389  uniwf  9794  rankuni  9838  rankc2  9846  rankxplim  9854  fin23lem17  10333  axcclem  10452  grurn  10797  istopon  23098  eltg3i  23147  cmpfi  23594  hmphdis  23982  ptcmpfi  23999  fbssfi  24023  mopnfss  24629  pliguhgr  30867  shsspwh  31627  circtopn  34250  hasheuni  34498  issgon  34536  sigaclci  34545  sigagenval  34554  dmsigagen  34558  imambfm  34676  bj-unirel  37720  salgenval  47068  salgenn0  47078  caragensspw  47256