MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sotr2 Structured version   Visualization version   GIF version

Theorem sotr2 5611
Description: A transitivity relation. (Read 𝐵𝐶 and 𝐶 < 𝐷 implies 𝐵 < 𝐷.) (Contributed by Mario Carneiro, 10-May-2013.)
Assertion
Ref Expression
sotr2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((¬ 𝐶𝑅𝐵𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Proof of Theorem sotr2
StepHypRef Expression
1 sotric 5607 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
21ancom2s 647 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
323adantr3 1168 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
43con2bid 354 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
5 breq1 5142 . . . . . 6 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
65biimpd 228 . . . . 5 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
76a1i 11 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
8 sotr 5603 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
98expd 415 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
107, 9jaod 856 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐶𝑅𝐷𝐵𝑅𝐷)))
114, 10sylbird 260 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐶𝑅𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
1211impd 410 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((¬ 𝐶𝑅𝐵𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5139   Or wor 5578
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-po 5579  df-so 5580
This theorem is referenced by:  nosupbnd1  27587  noinfbnd2  27604  slelttr  27630  erdszelem8  34706
  Copyright terms: Public domain W3C validator