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Theorem ordunel 7831
Description: The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
ordunel ((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ 𝐴)

Proof of Theorem ordunel
StepHypRef Expression
1 prssi 4826 . . 3 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
213adant1 1127 . 2 ((Ord 𝐴𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
3 ordelon 6395 . . . 4 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
433adant3 1129 . . 3 ((Ord 𝐴𝐵𝐴𝐶𝐴) → 𝐵 ∈ On)
5 ordelon 6395 . . 3 ((Ord 𝐴𝐶𝐴) → 𝐶 ∈ On)
6 ordunpr 7830 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
74, 5, 63imp3i2an 1342 . 2 ((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ {𝐵, 𝐶})
82, 7sseldd 3977 1 ((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084  wcel 2098  cun 3942  wss 3944  {cpr 4632  Ord word 6370  Oncon0 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375
This theorem is referenced by:  oaabs2  8670  dffi3  9461  unwf  9840  rankelun  9902  infxpenlem  10043  cfsmolem  10300  r1limwun  10766  wunex2  10768
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