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Theorem ssuncl 43560
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 3482 . . 3 𝑥 ∈ V
3 vex 3482 . . 3 𝑦 ∈ V
42, 3unex 7763 . 2 (𝑥𝑦) ∈ V
5 sseq1 4021 . 2 (𝑧 = (𝑥𝑦) → (𝑧𝐵 ↔ (𝑥𝑦) ⊆ 𝐵))
6 sseq1 4021 . 2 (𝑧 = 𝑥 → (𝑧𝐵𝑥𝐵))
7 sseq1 4021 . 2 (𝑧 = 𝑦 → (𝑧𝐵𝑦𝐵))
8 unss 4200 . . 3 ((𝑥𝐵𝑦𝐵) ↔ (𝑥𝑦) ⊆ 𝐵)
98biimpi 216 . 2 ((𝑥𝐵𝑦𝐵) → (𝑥𝑦) ⊆ 𝐵)
101, 4, 5, 6, 7, 9cllem0 43556 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  cun 3961  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by: (None)
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