Theorem ssuncl 43583

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

Hypothesis

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

Assertion

Ref	Expression
ssuncl	⊢ ∀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 ssuncl

Step	Hyp	Ref	Expression
1		ssficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}
2		vex 3484	. . 3 ⊢ 𝑥 ∈ V
3		vex 3484	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7764	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq1 4009	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵))
6		sseq1 4009	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
7		sseq1 4009	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
8		unss 4190	. . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)
9	8	biimpi 216	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
10	1, 4, 5, 6, 7, 9	cllem0 43579	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908

This theorem is referenced by: (None)