Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovord3 Structured version   Visualization version   GIF version

Theorem caovord3 6800
 Description: Ordering law. (Contributed by NM, 29-Feb-1996.)
Hypotheses
Ref Expression
caovord.1 𝐴 ∈ V
caovord.2 𝐵 ∈ V
caovord.3 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovord2.3 𝐶 ∈ V
caovord2.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caovord3.4 𝐷 ∈ V
Assertion
Ref Expression
caovord3 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovord3
StepHypRef Expression
1 caovord.1 . . . . 5 𝐴 ∈ V
2 caovord2.3 . . . . 5 𝐶 ∈ V
3 caovord.3 . . . . 5 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
4 caovord.2 . . . . 5 𝐵 ∈ V
5 caovord2.com . . . . 5 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
61, 2, 3, 4, 5caovord2 6799 . . . 4 (𝐵𝑆 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
76adantr 481 . . 3 ((𝐵𝑆𝐶𝑆) → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵)))
8 breq1 4616 . . 3 ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
97, 8sylan9bb 735 . 2 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
10 caovord3.4 . . . 4 𝐷 ∈ V
1110, 4, 3caovord 6798 . . 3 (𝐶𝑆 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
1211ad2antlr 762 . 2 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))
139, 12bitr4d 271 1 (((𝐵𝑆𝐶𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶𝐷𝑅𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   class class class wbr 4613  (class class class)co 6604 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-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-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607 This theorem is referenced by:  genpnnp  9771  ltsrpr  9842
 Copyright terms: Public domain W3C validator