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Theorem onun2i 6138
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 6127 . . 3 Ord 𝐵
3 on.1 . . . 4 𝐴 ∈ On
43onordi 6127 . . 3 Ord 𝐴
5 ordtri2or 6118 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴𝐴𝐵))
62, 4, 5mp2an 679 . 2 (𝐵𝐴𝐴𝐵)
73oneluni 6135 . . . 4 (𝐵𝐴 → (𝐴𝐵) = 𝐴)
87, 3syl6eqel 2868 . . 3 (𝐵𝐴 → (𝐴𝐵) ∈ On)
9 ssequn1 4038 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
10 eleq1 2847 . . . . 5 ((𝐴𝐵) = 𝐵 → ((𝐴𝐵) ∈ On ↔ 𝐵 ∈ On))
111, 10mpbiri 250 . . . 4 ((𝐴𝐵) = 𝐵 → (𝐴𝐵) ∈ On)
129, 11sylbi 209 . . 3 (𝐴𝐵 → (𝐴𝐵) ∈ On)
138, 12jaoi 843 . 2 ((𝐵𝐴𝐴𝐵) → (𝐴𝐵) ∈ On)
146, 13ax-mp 5 1 (𝐴𝐵) ∈ On
Colors of variables: wff setvar class
Syntax hints:  wo 833   = wceq 1507  wcel 2050  cun 3821  wss 3823  Ord word 6022  Oncon0 6023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-tr 5025  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-ord 6026  df-on 6027
This theorem is referenced by:  rankunb  9067  rankelun  9089  rankelpr  9090  inar1  9989
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