Theorem pwuni 4634

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 4627	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		selpw 4324	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
3	1, 2	sylibr 225	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
4	3	ssriv 3767	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2155   ⊆ wss 3734  𝒫 cpw 4317  ∪ cuni 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-ss 3748  df-pw 4319  df-uni 4597

This theorem is referenced by:  uniexr  7174  fipwuni  8543  uniwf  8901  rankuni  8945  rankc2  8953  rankxplim  8961  fin23lem17  9417  axcclem  9536  grurn  9880  istopon  21010  eltg3i  21059  cmpfi  21505  hmphdis  21893  ptcmpfi  21910  fbssfi  21934  mopnfss  22541  pliguhgr  27818  shsspwh  28580  circtopn  30372  hasheuni  30615  issgon  30654  sigaclci  30663  sigagenval  30671  dmsigagen  30675  imambfm  30792  salgenval  41202  salgenn0  41210  caragensspw  41387