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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) → (𝐵𝐶) ∈ {𝐵, 𝐶})

Proof of Theorem ordunpr
StepHypRef Expression
1 eloni 6332 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 eloni 6332 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
3 ordtri2or2 6421 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
41, 2, 3syl2an 597 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶𝐶𝐵))
54orcomd 870 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵𝐵𝐶))
6 ssequn2 4148 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
7 ssequn1 4145 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7orbi12i 914 . . 3 ((𝐶𝐵𝐵𝐶) ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
95, 8sylib 217 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
10 unexg 7688 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ V)
11 elprg 4612 . . 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 397  wo 846   = wceq 1542  wcel 2107  Vcvv 3448  cun 3913  wss 3915  {cpr 4593  Ord word 6321  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  ordunel  7767  r0weon  9955
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