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Theorem onsupcl3 42540
Description: The supremum of a set of ordinals is an ordinal. (Contributed by RP, 23-Jan-2025.)
Assertion
Ref Expression
onsupcl3 ((𝐴 ⊆ On ∧ 𝐴𝑉) → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} ∈ On)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem onsupcl3
StepHypRef Expression
1 onuniintrab 42533 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
2 ssonuni 7763 . . 3 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
32impcom 407 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 ∈ On)
41, 3eqeltrrd 2828 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3055  {crab 3426  wss 3943   cuni 4902   cint 4943  Oncon0 6357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361
This theorem is referenced by:  onsupex3  42541
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