Theorem ordun 6294

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 (𝐴 ∪ 𝐵))

Proof of Theorem ordun

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)
2		ordequn 6293	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵) → ((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵)))
3	1, 2	mpi 20	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵))
4		ordeq 6200	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐴))
5	4	biimprcd 252	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ∪ 𝐵) = 𝐴 → Ord (𝐴 ∪ 𝐵)))
6		ordeq 6200	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐵))
7	6	biimprcd 252	. . 3 ⊢ (Ord 𝐵 → ((𝐴 ∪ 𝐵) = 𝐵 → Ord (𝐴 ∪ 𝐵)))
8	5, 7	jaao 951	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (((𝐴 ∪ 𝐵) = 𝐴 ∨ (𝐴 ∪ 𝐵) = 𝐵) → Ord (𝐴 ∪ 𝐵)))
9	3, 8	mpd 15	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∪ cun 3936  Ord word 6192

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 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

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 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196

This theorem is referenced by:  ordsucun  7542  r0weon  9440