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Theorem onsupintrab 43219
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 43217 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → sup(𝐴, On, E ) = 𝐴)
2 onuniintrab 43214 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
31, 2eqtrd 2774 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 1536  wcel 2105  wral 3058  {crab 3432  wss 3962   cuni 4911   cint 4950   E cep 5587  Oncon0 6385  supcsup 9477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-iota 6515  df-riota 7387  df-sup 9479
This theorem is referenced by:  onsupintrab2  43220
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