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Theorem supex2g 33955
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 8555 . 2 sup(𝐵, 𝐴, 𝑅) = {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))}
2 rabexg 4972 . . 3 (𝐴𝐶 → {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V)
3 uniexg 7153 . . 3 ({𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V → {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V)
42, 3syl 17 . 2 (𝐴𝐶 {𝑥𝐴 ∣ (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))} ∈ V)
51, 4syl5eqel 2848 1 (𝐴𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350   cuni 4594   class class class wbr 4809  supcsup 8553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-rab 3064  df-v 3352  df-in 3739  df-ss 3746  df-uni 4595  df-sup 8555
This theorem is referenced by: (None)
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