Theorem unipwrVD 45139

Description: Virtual deduction proof of unipwr 45140. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unipwrVD	⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Proof of Theorem unipwrVD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3445	. . . . 5 ⊢ 𝑥 ∈ V
2	1	snid 4620	. . . 4 ⊢ 𝑥 ∈ {𝑥}
3		idn1 44882	. . . . 5 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		snelpwi 5393	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
5	3, 4	e1a 44935	. . . 4 ⊢ (   𝑥 ∈ 𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6		elunii 4869	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
7	2, 5, 6	e01an 45000	. . 3 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ ∪ 𝒫 𝐴   )
8	7	in1 44879	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	8	ssriv 3938	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 2114   ⊆ wss 3902  𝒫 cpw 4555  {csn 4581  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-un 3907  df-ss 3919  df-pw 4557  df-sn 4582  df-pr 4584  df-uni 4865  df-vd1 44878

This theorem is referenced by: (None)