Theorem unipwrVD 45641

Description: Virtual deduction proof of unipwr 45642. (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 3457	. . . . 5 ⊢ 𝑥 ∈ V
2	1	snid 4626	. . . 4 ⊢ 𝑥 ∈ {𝑥}
3		idn1 45384	. . . . 5 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		snelpwi 5423	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
5	3, 4	e1a 45437	. . . 4 ⊢ (   𝑥 ∈ 𝐴   ▶   {𝑥} ∈ 𝒫 𝐴   )
6		elunii 4875	. . . 4 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
7	2, 5, 6	e01an 45502	. . 3 ⊢ (   𝑥 ∈ 𝐴   ▶   𝑥 ∈ ∪ 𝒫 𝐴   )
8	7	in1 45381	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
9	8	ssriv 3938	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

This proof depends on syntax axioms:   ∈ wcel 2145   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-pw 4562  df-sn 4588  df-pr 4590  df-uni 4871  df-vd1 45380

This theorem is used by: (None)