Theorem unipwr 39721

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 5074. The proof of this theorem was automatically generated from unipwrVD 39720 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 3353	. . . 4 ⊢ 𝑥 ∈ V
2	1	snid 4366	. . 3 ⊢ 𝑥 ∈ {𝑥}
3		snelpwi 5068	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		elunii 4599	. . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
5	2, 3, 4	sylancr 581	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
6	5	ssriv 3765	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 2155   ⊆ wss 3732  𝒫 cpw 4315  {csn 4334  ∪ cuni 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-pw 4317  df-sn 4335  df-pr 4337  df-uni 4595

This theorem is referenced by: (None)