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Theorem ordun 6264
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 2820 . . 3 (𝐴𝐵) = (𝐴𝐵)
2 ordequn 6263 . . 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 6170 . . . 4 ((𝐴𝐵) = 𝐴 → (Ord (𝐴𝐵) ↔ Ord 𝐴))
54biimprcd 252 . . 3 (Ord 𝐴 → ((𝐴𝐵) = 𝐴 → Ord (𝐴𝐵)))
6 ordeq 6170 . . . 4 ((𝐴𝐵) = 𝐵 → (Ord (𝐴𝐵) ↔ Ord 𝐵))
76biimprcd 252 . . 3 (Ord 𝐵 → ((𝐴𝐵) = 𝐵 → Ord (𝐴𝐵)))
85, 7jaao 951 . 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 398  wo 843   = wceq 1537  cun 3907  Ord word 6162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-tr 5145  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-ord 6166
This theorem is referenced by:  ordsucun  7514  r0weon  9412
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