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

Theorem ordequn 5727
Description: The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordequn ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))

Proof of Theorem ordequn
StepHypRef Expression
1 ordtri2or2 5722 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
21orcomd 401 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵𝐵𝐶))
3 eqeq1 2609 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐵𝐶) = 𝐵))
4 ssequn2 3743 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
53, 4syl6rbbr 277 . . 3 (𝐴 = (𝐵𝐶) → (𝐶𝐵𝐴 = 𝐵))
6 eqeq1 2609 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐶 ↔ (𝐵𝐶) = 𝐶))
7 ssequn1 3740 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7syl6rbbr 277 . . 3 (𝐴 = (𝐵𝐶) → (𝐵𝐶𝐴 = 𝐶))
95, 8orbi12d 741 . 2 (𝐴 = (𝐵𝐶) → ((𝐶𝐵𝐵𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
102, 9syl5ibcom 233 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382   = wceq 1474  cun 3533  wss 3535  Ord word 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-tr 4671  df-eprel 4935  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-ord 5625
This theorem is referenced by:  ordun  5728  inar1  9449
  Copyright terms: Public domain W3C validator