Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ispos2 Structured version   Unicode version

Theorem ispos2 15903
 Description: A poset is an antisymmetric preset. EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
ispos2.b
ispos2.l
Assertion
Ref Expression
ispos2
Distinct variable groups:   ,,   ,,   , ,

Proof of Theorem ispos2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 3anan32 988 . . . . . . 7
21ralbii 2837 . . . . . 6
3 r19.26 2936 . . . . . 6
42, 3bitri 251 . . . . 5
542ralbii 2838 . . . 4
6 r19.26-2 2937 . . . . 5
7 rr19.3v 3193 . . . . . . 7
87ralbii 2837 . . . . . 6
98anbi2i 694 . . . . 5
106, 9bitri 251 . . . 4
115, 10bitri 251 . . 3
1211anbi2i 694 . 2
13 ispos2.b . . 3
14 ispos2.l . . 3
1513, 14ispos 15902 . 2
1613, 14isprs 15885 . . . 4
1716anbi1i 695 . . 3
18 anass 649 . . 3
1917, 18bitri 251 . 2
2012, 15, 193bitr4i 279 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 186   wa 369   w3a 976   wceq 1407   wcel 1844  wral 2756  cvv 3061   class class class wbr 4397  cfv 5571  cbs 14843  cple 14918   cpreset 15881  cpo 15895 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-preset 15883  df-poset 15901 This theorem is referenced by:  posprs  15904
 Copyright terms: Public domain W3C validator