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Theorem ordunpr 7835
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 6384 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 eloni 6384 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
3 ordtri2or2 6473 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
41, 2, 3syl2an 594 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶𝐶𝐵))
54orcomd 869 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵𝐵𝐶))
6 ssequn2 4185 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
7 ssequn1 4182 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7orbi12i 912 . . 3 ((𝐶𝐵𝐵𝐶) ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
95, 8sylib 217 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
10 unexg 7757 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ V)
11 elprg 4654 . . 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 256 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  Vcvv 3473  cun 3947  wss 3949  {cpr 4634  Ord word 6373  Oncon0 6374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378
This theorem is referenced by:  ordunel  7836  r0weon  10043
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