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Theorem ordun 6472
Description: The maximum (i.e., union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordun ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))

Proof of Theorem ordun
StepHypRef Expression
1 eqid 2726 . . 3 (𝐴𝐵) = (𝐴𝐵)
2 ordequn 6471 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) = (𝐴𝐵) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵)))
31, 2mpi 20 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵))
4 ordeq 6375 . . . 4 ((𝐴𝐵) = 𝐴 → (Ord (𝐴𝐵) ↔ Ord 𝐴))
54biimprcd 249 . . 3 (Ord 𝐴 → ((𝐴𝐵) = 𝐴 → Ord (𝐴𝐵)))
6 ordeq 6375 . . . 4 ((𝐴𝐵) = 𝐵 → (Ord (𝐴𝐵) ↔ Ord 𝐵))
76biimprcd 249 . . 3 (Ord 𝐵 → ((𝐴𝐵) = 𝐵 → Ord (𝐴𝐵)))
85, 7jaao 952 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (((𝐴𝐵) = 𝐴 ∨ (𝐴𝐵) = 𝐵) → Ord (𝐴𝐵)))
93, 8mpd 15 1 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1534  cun 3944  Ord word 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-tr 5263  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6371
This theorem is referenced by:  ordsucun  7826  r0weon  10048
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