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Theorem ordunel 6919
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 4293 . . 3 ((𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
213adant1 1072 . 2 ((Ord 𝐴𝐵𝐴𝐶𝐴) → {𝐵, 𝐶} ⊆ 𝐴)
3 ordelon 5664 . . . 4 ((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
433adant3 1074 . . 3 ((Ord 𝐴𝐵𝐴𝐶𝐴) → 𝐵 ∈ On)
5 ordelon 5664 . . . 4 ((Ord 𝐴𝐶𝐴) → 𝐶 ∈ On)
653adant2 1073 . . 3 ((Ord 𝐴𝐵𝐴𝐶𝐴) → 𝐶 ∈ On)
7 ordunpr 6918 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
84, 6, 7syl2anc 691 . 2 ((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ {𝐵, 𝐶})
92, 8sseldd 3569 1 ((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1031  wcel 1977  cun 3538  wss 3540  {cpr 4127  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-8 1979  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  ax-un 6847
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:  oaabs2  7612  dffi3  8220  unwf  8556  rankelun  8618  infxpenlem  8719  cfsmolem  8975  r1limwun  9437  wunex2  9439
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