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Theorem ordun 6043
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 2800 . . 3 (𝐴𝐵) = (𝐴𝐵)
2 ordequn 6042 . . 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 5949 . . . 4 ((𝐴𝐵) = 𝐴 → (Ord (𝐴𝐵) ↔ Ord 𝐴))
54biimprcd 242 . . 3 (Ord 𝐴 → ((𝐴𝐵) = 𝐴 → Ord (𝐴𝐵)))
6 ordeq 5949 . . . 4 ((𝐴𝐵) = 𝐵 → (Ord (𝐴𝐵) ↔ Ord 𝐵))
76biimprcd 242 . . 3 (Ord 𝐵 → ((𝐴𝐵) = 𝐵 → Ord (𝐴𝐵)))
85, 7jaao 978 . 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 385  wo 874   = wceq 1653  cun 3768  Ord word 5941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-tr 4947  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-ord 5945
This theorem is referenced by:  ordsucun  7260  r0weon  9122
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