Theorem superuncl 43924

Description: The class of all supersets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.)

Hypothesis

Ref	Expression
superficl.a	⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}

Assertion

Ref	Expression
superuncl	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem superuncl

Step	Hyp	Ref	Expression
1		superficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}
2		vex 3446	. . 3 ⊢ 𝑥 ∈ V
3		vex 3446	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7699	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq2 3962	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∪ 𝑦)))
6		sseq2 3962	. 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥))
7		sseq2 3962	. 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦))
8		ssun3 4134	. . 3 ⊢ (𝐵 ⊆ 𝑥 → 𝐵 ⊆ (𝑥 ∪ 𝑦))
9	8	adantr 480	. 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∪ 𝑦))
10	1, 4, 5, 6, 7, 9	cllem0 43922	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-uni 4866

This theorem is referenced by: (None)