Theorem ordunpr 7766

Description: The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
ordunpr	⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶})

Proof of Theorem ordunpr

Step	Hyp	Ref	Expression
1		eloni 6322	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		eloni 6322	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
3		ordtri2or2 6413	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
4	1, 2, 3	syl2an 597	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
5	4	orcomd 872	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶))
6		ssequn2 4120	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
7		ssequn1 4117	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
8	6, 7	orbi12i 915	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶) ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))
9	5, 8	sylib 218	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))
10		unexg 7686	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ V)
11		elprg 4580	. . 3 ⊢ ((𝐵 ∪ 𝐶) ∈ V → ((𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)))
12	10, 11	syl 17	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶)))
13	9, 12	mpbird 257	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3427   ∪ cun 3883   ⊆ wss 3885  {cpr 4559  Ord word 6311  Oncon0 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-tr 5182  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-ord 6315  df-on 6316

This theorem is referenced by:  ordunel  7767  r0weon  9923