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Theorem onun2i 5760
 Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
Hypotheses
Ref Expression
on.1 𝐴 ∈ On
on.2 𝐵 ∈ On
Assertion
Ref Expression
onun2i (𝐴𝐵) ∈ On

Proof of Theorem onun2i
StepHypRef Expression
1 on.2 . . . 4 𝐵 ∈ On
21onordi 5749 . . 3 Ord 𝐵
3 on.1 . . . 4 𝐴 ∈ On
43onordi 5749 . . 3 Ord 𝐴
5 ordtri2or 5739 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐴𝐵))
62, 4, 5mp2an 704 . 2 (𝐵𝐴𝐴𝐵)
73oneluni 5757 . . . 4 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
87, 3syl6eqel 2696 . . 3 (𝐵𝐴 → (𝐴𝐵) ∈ On)
9 ssequn1 3745 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
10 eleq1 2676 . . . . 5 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∈ On ↔ 𝐵 ∈ On))
111, 10mpbiri 247 . . . 4 ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On)
129, 11sylbi 206 . . 3 (𝐴𝐵 → (𝐴𝐵) ∈ On)
138, 12jaoi 393 . 2 ((𝐵𝐴𝐴𝐵) → (𝐴𝐵) ∈ On)
146, 13ax-mp 5 1 (𝐴𝐵) ∈ On
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  Ord word 5639  Oncon0 5640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644 This theorem is referenced by:  rankunb  8596  rankelun  8618  rankelpr  8619  inar1  9476
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