Theorem superuncl 42309

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 3478	. . 3 ⊢ 𝑥 ∈ V
3		vex 3478	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7732	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq2 4008	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ (𝑥 ∪ 𝑦)))
6		sseq2 4008	. 2 ⊢ (𝑧 = 𝑥 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑥))
7		sseq2 4008	. 2 ⊢ (𝑧 = 𝑦 → (𝐵 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑦))
8		ssun3 4174	. . 3 ⊢ (𝐵 ⊆ 𝑥 → 𝐵 ⊆ (𝑥 ∪ 𝑦))
9	8	adantr 481	. 2 ⊢ ((𝐵 ⊆ 𝑥 ∧ 𝐵 ⊆ 𝑦) → 𝐵 ⊆ (𝑥 ∪ 𝑦))
10	1, 4, 5, 6, 7, 9	cllem0 42307	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  Vcvv 3474   ∪ cun 3946   ⊆ wss 3948

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909

This theorem is referenced by: (None)