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Theorem ordequn 6489
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 6485 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
21orcomd 871 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵𝐵𝐶))
3 ssequn2 4199 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
4 eqeq1 2739 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐵𝐶) = 𝐵))
53, 4bitr4id 290 . . 3 (𝐴 = (𝐵𝐶) → (𝐶𝐵𝐴 = 𝐵))
6 ssequn1 4196 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
7 eqeq1 2739 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐶 ↔ (𝐵𝐶) = 𝐶))
86, 7bitr4id 290 . . 3 (𝐴 = (𝐵𝐶) → (𝐵𝐶𝐴 = 𝐶))
95, 8orbi12d 918 . 2 (𝐴 = (𝐵𝐶) → ((𝐶𝐵𝐵𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
102, 9syl5ibcom 245 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  cun 3961  wss 3963  Ord word 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389
This theorem is referenced by:  ordun  6490  inar1  10813
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