Theorem ordunpr 7173

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 5876	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		eloni 5876	. . . . 5 ⊢ (𝐶 ∈ On → Ord 𝐶)
3		ordtri2or2 5966	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
4	1, 2, 3	syl2an 583	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
5	4	orcomd 860	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶))
6		ssequn2 3937	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
7		ssequn1 3934	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
8	6, 7	orbi12i 900	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐶) ↔ ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))
9	5, 8	sylib 208	. 2 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵 ∪ 𝐶) = 𝐵 ∨ (𝐵 ∪ 𝐶) = 𝐶))
10		unexg 7106	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ V)
11		elprg 4336	. . 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 247	1 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∨ wo 836   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∪ cun 3721   ⊆ wss 3723  {cpr 4318  Ord word 5865  Oncon0 5866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870

This theorem is referenced by:  ordunel  7174  r0weon  9035