Theorem ssuncl 43000

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 3475	. . 3 ⊢ 𝑥 ∈ V
3		vex 3475	. . 3 ⊢ 𝑦 ∈ V
4	2, 3	unex 7748	. 2 ⊢ (𝑥 ∪ 𝑦) ∈ V
5		sseq1 4005	. 2 ⊢ (𝑧 = (𝑥 ∪ 𝑦) → (𝑧 ⊆ 𝐵 ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵))
6		sseq1 4005	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝐵 ↔ 𝑥 ⊆ 𝐵))
7		sseq1 4005	. 2 ⊢ (𝑧 = 𝑦 → (𝑧 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵))
8		unss 4184	. . 3 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) ↔ (𝑥 ∪ 𝑦) ⊆ 𝐵)
9	8	biimpi 215	. 2 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝐵) → (𝑥 ∪ 𝑦) ⊆ 𝐵)
10	1, 4, 5, 6, 7, 9	cllem0 42996	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2705  ∀wral 3058  Vcvv 3471   ∪ cun 3945   ⊆ wss 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4909

This theorem is referenced by: (None)