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Theorem onun2 6276
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 4070 . . 3 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
2 eleq1a 2828 . . . 4 (𝐵 ∈ On → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
32adantl 485 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On))
41, 3syl5bi 245 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐴𝐵) ∈ On))
5 ssequn2 4073 . . 3 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐴)
6 eleq1a 2828 . . . 4 (𝐴 ∈ On → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
76adantr 484 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ∈ On))
85, 7syl5bi 245 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵𝐴 → (𝐴𝐵) ∈ On))
9 eloni 6182 . . 3 (𝐴 ∈ On → Ord 𝐴)
10 eloni 6182 . . 3 (𝐵 ∈ On → Ord 𝐵)
11 ordtri2or2 6268 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
129, 10, 11syl2an 599 . 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 399  wo 846   = wceq 1542  wcel 2114  cun 3841  wss 3843  Ord word 6171  Oncon0 6172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-ord 6175  df-on 6176
This theorem is referenced by:  onun2i  6288  nosupinfsep  33578
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