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Theorem onun2 6468
Description: The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)
Assertion
Ref Expression
onun2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)

Proof of Theorem onun2
StepHypRef Expression
1 ssequn1 4178 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2829 . . . 4 (𝐵 ∈ On → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
32adantl 483 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
41, 3biimtrid 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴𝐵) ∈ On))
5 ssequn2 4181 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2829 . . . 4 (𝐴 ∈ On → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
76adantr 482 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
85, 7biimtrid 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → (𝐴𝐵) ∈ On))
9 eloni 6370 . . 3 (𝐴 ∈ On → Ord 𝐴)
10 eloni 6370 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordtri2or2 6459 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 597 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 859 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  cun 3944  wss 3946  Ord word 6359  Oncon0 6360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-tr 5264  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6363  df-on 6364
This theorem is referenced by:  onun2i  6482  nosupinfsep  27214  onexlimgt  41924  omabs2  42014  onsucunitp  42055
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