Theorem unipwrVD 44857

Description: Virtual deduction proof of unipwr 44858. (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 3483	. . . . 5 ⊢ 𝑥 ∈ V
2	1	snid 4661	. . . 4 ⊢ 𝑥 ∈ {𝑥}
3		idn1 44599	. . . . 5 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		snelpwi 5447	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
5	3, 4	e1a 44652	. . . 4 ⊢ (   𝑥 ∈ 𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6		elunii 4911	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
7	2, 5, 6	e01an 44717	. . 3 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ ∪ 𝒫 𝐴   )
8	7	in1 44596	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	8	ssriv 3986	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 2107   ⊆ wss 3950  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906

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 2707  ax-sep 5295  ax-pr 5431

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 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907  df-vd1 44595

This theorem is referenced by: (None)