Theorem ssuncl 44147

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 3459	. . 3 ⊢ 𝑥 ∈ V
3		vex 3459	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7728	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq1 3962	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵))
6		sseq1 3962	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
7		sseq1 3962	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
8		unss 4143	. . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)
9	8	biimpi 218	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
10	1, 4, 5, 6, 7, 9	cllem0 44143	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Syntax hints:   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {cab 2741  ∀wral 3077  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-v 3457  df-un 3910  df-ss 3922  df-sn 4584  df-pr 4586  df-uni 4867

This theorem is referenced by: (None)