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Theorem ordequn 6487
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 6483 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
21orcomd 872 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵𝐵𝐶))
3 ssequn2 4189 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
4 eqeq1 2741 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐵𝐶) = 𝐵))
53, 4bitr4id 290 . . 3 (𝐴 = (𝐵𝐶) → (𝐶𝐵𝐴 = 𝐵))
6 ssequn1 4186 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
7 eqeq1 2741 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐶 ↔ (𝐵𝐶) = 𝐶))
86, 7bitr4id 290 . . 3 (𝐴 = (𝐵𝐶) → (𝐵𝐶𝐴 = 𝐶))
95, 8orbi12d 919 . 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 848   = wceq 1540  cun 3949  wss 3951  Ord word 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387
This theorem is referenced by:  ordun  6488  inar1  10815
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