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Theorem supex2g 35120
Description: Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
supex2g (𝐴𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Proof of Theorem supex2g
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sup 8903 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rabexg 5220 . . 3 (𝐴𝐶 → {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V)
32uniexd 7462 . 2 (𝐴𝐶 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V)
41, 3eqeltrid 2920 1 (𝐴𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2115  wral 3133  wrex 3134  {crab 3137  Vcvv 3480   cuni 4824   class class class wbr 5052  supcsup 8901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-uni 4825  df-sup 8903
This theorem is referenced by: (None)
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