Theorem pwuni 4255

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 3895	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		vex 2782	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 3635	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
4	1, 3	sylibr 134	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
5	4	ssriv 3208	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2180   ⊆ wss 3177  𝒫 cpw 3629  ∪ cuni 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631  df-uni 3868

This theorem is referenced by:  uniexb  4541  2pwuninelg  6399  istopon  14652  eltg3i  14695  mopnfss  15086