Theorem unipwr 38090

Description: A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4845. The proof of this theorem was automatically generated from unipwrVD 38089 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 3176	. . . 4 ⊢ 𝑥 ∈ V
2	1	snid 4155	. . 3 ⊢ 𝑥 ∈ {𝑥}
3		snelpwi 4839	. . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 𝐴)
4		elunii 4377	. . 3 ⊢ ((𝑥 ∈ {𝑥} ∧ {𝑥} ∈ 𝒫 𝐴) → 𝑥 ∈ ∪ 𝒫 𝐴)
5	2, 3, 4	sylancr 694	. 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ∪ 𝒫 𝐴)
6	5	ssriv 3572	1 ⊢ 𝐴 ⊆ ∪ 𝒫 𝐴

Syntax hints:   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by: (None)