Theorem unipwr 45814

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5418. The proof of this theorem was automatically generated from unipwrVD 45813 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 3455	. . . 4 ⊢ 𝑥 ∈ V
2	1	snid 4623	. . 3 ⊢ 𝑥 ∈ {𝑥}
3		snelpwi 5412	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		elunii 4872	. . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
5	2, 3, 4	sylancr 599	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
6	5	ssriv 3935	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

This proof depends on syntax axioms:   ∈ wcel 2145   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867

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 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by: (None)