Theorem unipwrVD 45568

Description: Virtual deduction proof of unipwr 45569. (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 3458	. . . . 5 ⊢ 𝑥 ∈ V
2	1	snid 4627	. . . 4 ⊢ 𝑥 ∈ {𝑥}
3		idn1 45311	. . . . 5 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		snelpwi 5424	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
5	3, 4	e1a 45364	. . . 4 ⊢ (   𝑥 ∈ 𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6		elunii 4876	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
7	2, 5, 6	e01an 45429	. . 3 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ ∪ 𝒫 𝐴   )
8	7	in1 45308	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	8	ssriv 3940	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

This proof depends on syntax axioms:   ∈ wcel 2142   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588  ∪ cuni 4871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909  df-ss 3921  df-pw 4563  df-sn 4589  df-pr 4591  df-uni 4872  df-vd1 45307

This theorem is used by: (None)