Theorem unipwr 44810

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5435. The proof of this theorem was automatically generated from unipwrVD 44809 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unipwr	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Proof of Theorem unipwr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3467	. . . 4 ⊢ 𝑥 ∈ V
2	1	snid 4642	. . 3 ⊢ 𝑥 ∈ {𝑥}
3		snelpwi 5428	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		elunii 4892	. . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
5	2, 3, 4	sylancr 587	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
6	5	ssriv 3967	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 2107   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ∪ cuni 4887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3465  df-un 3936  df-ss 3948  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4888

This theorem is referenced by: (None)