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Theorem dmdi2 28547
Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)
Assertion
Ref Expression
dmdi2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))

Proof of Theorem dmdi2
StepHypRef Expression
1 dmdi 28545 . 2 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → ((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)))
2 eqimss2 3621 . 2 (((𝐶𝐴) ∨ 𝐵) = (𝐶 ∩ (𝐴 𝐵)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
31, 2syl 17 1 (((𝐴C𝐵C𝐶C ) ∧ (𝐴 𝑀* 𝐵𝐵𝐶)) → (𝐶 ∩ (𝐴 𝐵)) ⊆ ((𝐶𝐴) ∨ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  cin 3539  wss 3540   class class class wbr 4583  (class class class)co 6549   C cch 27170   chj 27174   𝑀* cdmd 27208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-dmd 28524
This theorem is referenced by: (None)
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