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Theorem onun2 6492
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 4186 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2836 . . . 4 (𝐵 ∈ On → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
32adantl 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
41, 3biimtrid 242 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴𝐵) ∈ On))
5 ssequn2 4189 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2836 . . . 4 (𝐴 ∈ On → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
76adantr 480 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
85, 7biimtrid 242 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → (𝐴𝐵) ∈ On))
9 eloni 6394 . . 3 (𝐴 ∈ On → Ord 𝐴)
10 eloni 6394 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordtri2or2 6483 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 861 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  cun 3949  wss 3951  Ord word 6383  Oncon0 6384
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-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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  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-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  onun2i  6506  nosupinfsep  27777  onexlimgt  43255  omabs2  43345  onsucunitp  43386
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