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Theorem superuncl 36892
Description: The class of all supersets of a class is closed under binary union. (Contributed by Richard Penner, 3-Jan-2020.)
Hypothesis
Ref Expression
superficl.a 𝐴 = {𝑧𝐵𝑧}
Assertion
Ref Expression
superuncl 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 superuncl
StepHypRef Expression
1 superficl.a . 2 𝐴 = {𝑧𝐵𝑧}
2 vex 3176 . . 3 𝑥 ∈ V
3 vex 3176 . . 3 𝑦 ∈ V
42, 3unex 6854 . 2 (𝑥𝑦) ∈ V
5 sseq2 3590 . 2 (𝑧 = (𝑥𝑦) → (𝐵𝑧𝐵 ⊆ (𝑥𝑦)))
6 sseq2 3590 . 2 (𝑧 = 𝑥 → (𝐵𝑧𝐵𝑥))
7 sseq2 3590 . 2 (𝑧 = 𝑦 → (𝐵𝑧𝐵𝑦))
8 ssun3 3740 . . 3 (𝐵𝑥𝐵 ⊆ (𝑥𝑦))
98adantr 480 . 2 ((𝐵𝑥𝐵𝑦) → 𝐵 ⊆ (𝑥𝑦))
101, 4, 5, 6, 7, 9cllem0 36890 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  {cab 2596  wral 2896  Vcvv 3173  cun 3538  wss 3540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373
This theorem is referenced by: (None)
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