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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) → (𝐵𝐶) ∈ {𝐵, 𝐶})

Proof of Theorem ordunpr
StepHypRef 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) → (𝐵𝐶𝐶𝐵))
41, 2, 3syl2an 583 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶𝐶𝐵))
54orcomd 860 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵𝐵𝐶))
6 ssequn2 3937 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
7 ssequn1 3934 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7orbi12i 900 . . 3 ((𝐶𝐵𝐵𝐶) ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
95, 8sylib 208 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
10 unexg 7106 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ V)
11 elprg 4336 . . 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 247 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
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
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