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Theorem sotr2 5029
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 5026 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
21ancom2s 843 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
323adantr3 1220 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
43con2bid 344 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
5 breq1 4621 . . . . . 6 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
65biimpd 219 . . . . 5 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
76a1i 11 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
8 sotr 5022 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
98expd 452 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
107, 9jaod 395 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐶𝑅𝐷𝐵𝑅𝐷)))
114, 10sylbird 250 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐶𝑅𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
1211impd 447 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((¬ 𝐶𝑅𝐵𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618   Or wor 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-po 5000  df-so 5001
This theorem is referenced by:  erdszelem8  30923
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