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Theorem onsupintrab 43243
Description: The supremum of a set of ordinals is the intersection of every ordinal greater-than-or-equal to every member of the set. Definition 2.9 of [Schloeder] p. 5. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
onsupintrab ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem onsupintrab
StepHypRef Expression
1 onsupuni 43241 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)
2 onuniintrab 43238 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
31, 2eqtrd 2777 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951   cuni 4907   cint 4946   E cep 5583  Oncon0 6384  supcsup 9480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-iota 6514  df-riota 7388  df-sup 9482
This theorem is referenced by:  onsupintrab2  43244
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