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Theorem ordun 5990
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 2760 . . 3 (𝐴𝐵) = (𝐴𝐵)
2 ordequn 5989 . . 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 5891 . . . 4 ((𝐴𝐵) = 𝐴 → (Ord (𝐴𝐵) ↔ Ord 𝐴))
54biimprcd 240 . . 3 (Ord 𝐴 → ((𝐴𝐵) = 𝐴 → Ord (𝐴𝐵)))
6 ordeq 5891 . . . 4 ((𝐴𝐵) = 𝐵 → (Ord (𝐴𝐵) ↔ Ord 𝐵))
76biimprcd 240 . . 3 (Ord 𝐵 → ((𝐴𝐵) = 𝐵 → Ord (𝐴𝐵)))
85, 7jaao 532 . 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  wo 382  wa 383   = wceq 1632  cun 3713  Ord word 5883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-tr 4905  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-ord 5887
This theorem is referenced by:  ordsucun  7190  r0weon  9025
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