Theorem ordun 6488

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 2737	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)
2		ordequn 6487	. . 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 6391	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐴))
5	4	biimprcd 250	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ∪ 𝐵) = 𝐴 → Ord (𝐴 ∪ 𝐵)))
6		ordeq 6391	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐵))
7	6	biimprcd 250	. . 3 ⊢ (Ord 𝐵 → ((𝐴 ∪ 𝐵) = 𝐵 → Ord (𝐴 ∪ 𝐵)))
8	5, 7	jaao 957	. 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 395   ∨ wo 848   = wceq 1540   ∪ cun 3949  Ord word 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387

This theorem is referenced by:  ordsucun  7845  r0weon  10052