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Theorem sotr2 5498
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 5494 . . . . . 6 ((𝑅 Or 𝐴 ∧ (𝐶𝐴𝐵𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
21ancom2s 646 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
323adantr3 1163 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵𝐵𝑅𝐶)))
43con2bid 356 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
5 breq1 5060 . . . . . 6 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
65biimpd 230 . . . . 5 (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷))
76a1i 11 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐶 = 𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
8 sotr 5490 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))
98expd 416 . . . 4 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (𝐵𝑅𝐶 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
107, 9jaod 853 . . 3 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((𝐶 = 𝐵𝐵𝑅𝐶) → (𝐶𝑅𝐷𝐵𝑅𝐷)))
114, 10sylbird 261 . 2 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → (¬ 𝐶𝑅𝐵 → (𝐶𝑅𝐷𝐵𝑅𝐷)))
1211impd 411 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ((¬ 𝐶𝑅𝐵𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057   Or wor 5466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-po 5467  df-so 5468
This theorem is referenced by:  erdszelem8  32342  nosupbnd1  33111  slelttr  33133
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