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Theorem oninfunirab 43249
Description: The infimum of a non-empty class of ordinals is the union of every ordinal less-than-or-equal to every element of that class. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
oninfunirab ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem oninfunirab
StepHypRef Expression
1 oninfint 43248 . 2 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = 𝐴)
2 onintunirab 43239 . 2 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
31, 2eqtrd 2777 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → inf(𝐴, On, E ) = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑥𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wne 2940  wral 3061  {crab 3436  wss 3951  c0 4333   cuni 4907   cint 4946   E cep 5583  Oncon0 6384  infcinf 9481
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
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-cnv 5693  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-riota 7388  df-sup 9482  df-inf 9483
This theorem is referenced by: (None)
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