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Theorem onun2 6370
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 4114 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2834 . . . 4 (𝐵 ∈ On → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
32adantl 482 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
41, 3syl5bi 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴𝐵) ∈ On))
5 ssequn2 4117 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2834 . . . 4 (𝐴 ∈ On → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
76adantr 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
85, 7syl5bi 241 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → (𝐴𝐵) ∈ On))
9 eloni 6276 . . 3 (𝐴 ∈ On → Ord 𝐴)
10 eloni 6276 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordtri2or2 6362 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 596 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐵𝐴))
134, 8, 12mpjaod 857 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 844   = wceq 1539  wcel 2106  cun 3885  wss 3887  Ord word 6265  Oncon0 6266
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270
This theorem is referenced by:  onun2i  6382  nosupinfsep  33935
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