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Theorem ordunpr 7821
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) → (𝐵𝐶) ∈ {𝐵, 𝐶})

Proof of Theorem ordunpr
StepHypRef Expression
1 eloni 6373 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 eloni 6373 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
3 ordtri2or2 6462 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
41, 2, 3syl2an 595 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶𝐶𝐵))
54orcomd 870 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵𝐵𝐶))
6 ssequn2 4179 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
7 ssequn1 4176 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7orbi12i 913 . . 3 ((𝐶𝐵𝐵𝐶) ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
95, 8sylib 217 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
10 unexg 7743 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ V)
11 elprg 4645 . . 3 ((𝐵𝐶) ∈ V → ((𝐵𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶)))
1210, 11syl 17 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) ∈ {𝐵, 𝐶} ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶)))
139, 12mpbird 257 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  Vcvv 3469  cun 3942  wss 3944  {cpr 4626  Ord word 6362  Oncon0 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367
This theorem is referenced by:  ordunel  7822  r0weon  10021
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