Theorem ordunmax 35792

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 6482	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
2	1	ord 878	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → 𝐵 ⊆ 𝐴))
3		ssequn2 4190	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
4	2, 3	imbitrdi 254	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐴))
5		iffalse 4494	. . . 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 2785	. . 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 4187	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
10	9	biimpi 219	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) = 𝐵)
11		iftrue 4491	. . 3 ⊢ (𝐴 ⊆ 𝐵 → if(𝐴 ⊆ 𝐵, 𝐵, 𝐴) = 𝐵)
12	10, 11	eqtr4d 2801	. 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 1570   ⊆ wss 3900   ∪ cun 4152  ifcif 4485  Ord word 6379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383

This theorem is used by: (None)