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Theorem ordequn 6284
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 6280 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
21orcomd 867 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐶𝐵𝐵𝐶))
3 eqeq1 2823 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵 ↔ (𝐵𝐶) = 𝐵))
4 ssequn2 4157 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
53, 4syl6rbbr 292 . . 3 (𝐴 = (𝐵𝐶) → (𝐶𝐵𝐴 = 𝐵))
6 eqeq1 2823 . . . 4 (𝐴 = (𝐵𝐶) → (𝐴 = 𝐶 ↔ (𝐵𝐶) = 𝐶))
7 ssequn1 4154 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
86, 7syl6rbbr 292 . . 3 (𝐴 = (𝐵𝐶) → (𝐵𝐶𝐴 = 𝐶))
95, 8orbi12d 914 . 2 (𝐴 = (𝐵𝐶) → ((𝐶𝐵𝐵𝐶) ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
102, 9syl5ibcom 247 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐵𝐶) → (𝐴 = 𝐵𝐴 = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1530  cun 3932  wss 3934  Ord word 6183
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
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
This theorem is referenced by:  ordun  6285  inar1  10189
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