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Theorem onsupcl3 42692
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 42685 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
2 ssonuni 7788 . . 3 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
32impcom 406 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → 𝐴 ∈ On)
41, 3eqeltrrd 2830 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  wral 3058  {crab 3430  wss 3949   cuni 4912   cint 4953  Oncon0 6374
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378
This theorem is referenced by:  onsupex3  42693
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