Theorem unipwr 43429

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5444. The proof of this theorem was automatically generated from unipwrVD 43428 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 3478	. . . 4 ⊢ 𝑥 ∈ V
2	1	snid 4659	. . 3 ⊢ 𝑥 ∈ {𝑥}
3		snelpwi 5437	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		elunii 4907	. . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
5	2, 3, 4	sylancr 587	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
6	5	ssriv 3983	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 2106   ⊆ wss 3945  𝒫 cpw 4597  {csn 4623  ∪ cuni 4902

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5293  ax-pr 5421

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3950  df-in 3952  df-ss 3962  df-pw 4599  df-sn 4624  df-pr 4626  df-uni 4903

This theorem is referenced by: (None)