Theorem ordun 6499

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 2740	. . 3 ⊢ (𝐴 ∪ 𝐵) = (𝐴 ∪ 𝐵)
2		ordequn 6498	. . 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 6402	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → (Ord (𝐴 ∪ 𝐵) ↔ Ord 𝐴))
5	4	biimprcd 250	. . 3 ⊢ (Ord 𝐴 → ((𝐴 ∪ 𝐵) = 𝐴 → Ord (𝐴 ∪ 𝐵)))
6		ordeq 6402	. . . 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 955	. 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 846   = wceq 1537   ∪ cun 3974  Ord word 6394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398

This theorem is referenced by:  ordsucun  7861  r0weon  10081