MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordunpr Structured version   Visualization version   GIF version

Theorem ordunpr 7533
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 6194 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 eloni 6194 . . . . 5 (𝐶 ∈ On → Ord 𝐶)
3 ordtri2or2 6280 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
41, 2, 3syl2an 597 . . . 4 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶𝐶𝐵))
54orcomd 867 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶𝐵𝐵𝐶))
6 ssequn2 4157 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
7 ssequn1 4154 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7orbi12i 910 . . 3 ((𝐶𝐵𝐵𝐶) ↔ ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
95, 8sylib 220 . 2 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐵𝐶) = 𝐵 ∨ (𝐵𝐶) = 𝐶))
10 unexg 7464 . . 3 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ V)
11 elprg 4580 . . 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 259 1 ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1530  wcel 2107  Vcvv 3493  cun 3932  wss 3934  {cpr 4561  Ord word 6183  Oncon0 6184
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  ordunel  7534  r0weon  9430
  Copyright terms: Public domain W3C validator