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Theorem ssuncl 41747
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 = {𝑧𝑧𝐵}
Assertion
Ref Expression
ssuncl 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴   𝑧,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥,𝑦)

Proof of Theorem ssuncl
StepHypRef Expression
1 ssficl.a . 2 𝐴 = {𝑧𝑧𝐵}
2 vex 3448 . . 3 𝑥 ∈ V
3 vex 3448 . . 3 𝑦 ∈ V
42, 3unex 7673 . 2 (𝑥𝑦) ∈ V
5 sseq1 3968 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
6 sseq1 3968 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
7 sseq1 3968 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
8 unss 4143 . . 3 ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝑦) ⊆ 𝐵)
98biimpi 215 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
101, 4, 5, 6, 7, 9cllem0 41743 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  {cab 2715  wral 3063  Vcvv 3444  cun 3907  wss 3909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-sn 4586  df-pr 4588  df-uni 4865
This theorem is referenced by: (None)
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