Theorem ordunpr 7770

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 6328	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		eloni 6328	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
3		ordtri2or2 6419	. . . . 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 4142	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
7		ssequn1 4139	. . . 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 7690	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ V)
11		elprg 4604	. . 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 3441   ∪ cun 3900   ⊆ wss 3902  {cpr 4583  Ord word 6317  Oncon0 6318

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 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322

This theorem is referenced by:  ordunel  7771  r0weon  9926