Theorem superuncl 44354

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 3461	. . 3 ⊢ 𝑥 ∈ V
3		vex 3461	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7752	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq2 3964	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∪ 𝑦)))
6		sseq2 3964	. 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥))
7		sseq2 3964	. 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦))
8		ssun3 4133	. . 3 ⊢ (𝐵 ⊆ 𝑥 → 𝐵 ⊆ (𝑥 ∪ 𝑦))
9	8	adantr 486	. 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∪ 𝑦))
10	1, 4, 5, 6, 7, 9	cllem0 44352	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  Vcvv 3457   ∪ cun 3904   ⊆ wss 3906

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-un 3911  df-ss 3923  df-sn 4592  df-pr 4594  df-uni 4875

This theorem is used by: (None)