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Theorem ordequn 5816
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 5811 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
21orcomd 403 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵𝐵𝐶))
3 eqeq1 2624 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐵𝐶) = 𝐵))
4 ssequn2 3778 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
53, 4syl6rbbr 279 . . 3 (𝐴 = (𝐵𝐶) → (𝐶𝐵𝐴 = 𝐵))
6 eqeq1 2624 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐶 ↔ (𝐵𝐶) = 𝐶))
7 ssequn1 3775 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7syl6rbbr 279 . . 3 (𝐴 = (𝐵𝐶) → (𝐵𝐶𝐴 = 𝐶))
95, 8orbi12d 745 . 2 (𝐴 = (𝐵𝐶) → ((𝐶𝐵𝐵𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
102, 9syl5ibcom 235 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1481  cun 3565  wss 3567  Ord word 5710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714
This theorem is referenced by:  ordun  5817  inar1  9582
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