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Theorem superuncl 37389
 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 3192 . . 3 𝑥 ∈ V
3 vex 3192 . . 3 𝑦 ∈ V
42, 3unex 6916 . 2 (𝑥𝑦) ∈ V
5 sseq2 3611 . 2 (𝑧 = (𝑥𝑦) → (𝐵𝑧𝐵 ⊆ (𝑥𝑦)))
6 sseq2 3611 . 2 (𝑧 = 𝑥 → (𝐵𝑧𝐵𝑥))
7 sseq2 3611 . 2 (𝑧 = 𝑦 → (𝐵𝑧𝐵𝑦))
8 ssun3 3761 . . 3 (𝐵𝑥𝐵 ⊆ (𝑥𝑦))
98adantr 481 . 2 ((𝐵𝑥𝐵𝑦) → 𝐵 ⊆ (𝑥𝑦))
101, 4, 5, 6, 7, 9cllem0 37387 1 𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480   ∈ wcel 1987  {cab 2607  ∀wral 2907  Vcvv 3189   ∪ cun 3557   ⊆ wss 3559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-sn 4154  df-pr 4156  df-uni 4408 This theorem is referenced by: (None)
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