Theorem ordunmax 35758

Description: The union of two ordinals is equal to their maximum. (Contributed by BTernaryTau, 7-Oct-2026.)

Assertion

Ref	Expression
ordunmax	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴))

Proof of Theorem ordunmax

Step	Hyp	Ref	Expression
1		ordtri2or2 6464	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
2	1	ord 878	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → 𝐵 ⊆ 𝐴))
3		ssequn2 4187	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
4	2, 3	imbitrdi 254	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐴))
5		iffalse 4490	. . . 4 ⊢ (¬ 𝐴 ⊆ 𝐵 → if(𝐴 ⊆ 𝐵, 𝐵, 𝐴) = 𝐴)
6	4, 5	jca2 523	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → ((𝐴 ∪ 𝐵) = 𝐴 ∧ if(𝐴 ⊆ 𝐵, 𝐵, 𝐴) = 𝐴)))
7		eqtr3 2782	. . 3 ⊢ (((𝐴 ∪ 𝐵) = 𝐴 ∧ if(𝐴 ⊆ 𝐵, 𝐵, 𝐴) = 𝐴) → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴))
8	6, 7	syl6 36	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴)))
9		ssequn1 4184	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
10	9	biimpi 219	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
11		iftrue 4487	. . 3 ⊢ (𝐴 ⊆ 𝐵 → if(𝐴 ⊆ 𝐵, 𝐵, 𝐴) = 𝐵)
12	10, 11	eqtr4d 2798	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴))
13	8, 12	pm2.61d2 183	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∪ 𝐵) = if(𝐴 ⊆ 𝐵, 𝐵, 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ⊆ wss 3895   ∪ cun 4149  ifcif 4481  Ord word 6361

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365

This theorem is used by: (None)