Theorem pwuni 4165

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 3811	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
2		vex 2724	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 3559	. . 3 ⊢ (𝑥 ∈ 𝒫 ∪ 𝐴 ↔ 𝑥 ⊆ ∪ 𝐴)
4	1, 3	sylibr 133	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 ∪ 𝐴)
5	4	ssriv 3141	1 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴

Syntax hints:   ∈ wcel 2135   ⊆ wss 3111  𝒫 cpw 3553  ∪ cuni 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117  df-ss 3124  df-pw 3555  df-uni 3784

This theorem is referenced by:  uniexb  4445  2pwuninelg  6242  istopon  12552  eltg3i  12597  mopnfss  12988