Theorem pwuni 4474

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 4467	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		selpw 4165	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
3	1, 2	sylibr 224	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
4	3	ssriv 3607	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 1990   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  uniexr  6972  fipwuni  8332  uniwf  8682  rankuni  8726  rankc2  8734  rankxplim  8742  fin23lem17  9160  axcclem  9279  grurn  9623  istopon  20717  eltg3i  20765  cmpfi  21211  hmphdis  21599  ptcmpfi  21616  fbssfi  21641  mopnfss  22248  pliguhgr  27338  shsspwh  28103  circtopn  29904  hasheuni  30147  issgon  30186  sigaclci  30195  sigagenval  30203  dmsigagen  30207  imambfm  30324  salgenval  40541  salgenn0  40549  caragensspw  40723