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Theorem onun2 6494
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 4196 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2834 . . . 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 4199 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2834 . . . 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 6396 . . 3 (𝐴 ∈ On → Ord 𝐴)
10 eloni 6396 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordtri2or2 6485 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 860 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  cun 3961  wss 3963  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  onun2i  6508  nosupinfsep  27792  onexlimgt  43232  omabs2  43322  onsucunitp  43363
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