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Theorem ispos2 14397
 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 948 . . . . . . 7
21ralbii 2721 . . . . . 6
3 r19.26 2830 . . . . . 6
42, 3bitri 241 . . . . 5
542ralbii 2723 . . . 4
6 r19.26-2 2831 . . . . 5
7 rr19.3v 3069 . . . . . . 7
87ralbii 2721 . . . . . 6
98anbi2i 676 . . . . 5
106, 9bitri 241 . . . 4
115, 10bitri 241 . . 3
1211anbi2i 676 . 2
13 ispos2.b . . 3
14 ispos2.l . . 3
1513, 14ispos 14396 . 2
1613, 14isprs 14379 . . . 4
1716anbi1i 677 . . 3
18 anass 631 . . 3
1917, 18bitri 241 . 2
2012, 15, 193bitr4i 269 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  cvv 2948   class class class wbr 4204  cfv 5446  cbs 13461  cple 13528   cpreset 14375  cpo 14389 This theorem is referenced by:  posprs  14398 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-preset 14377  df-poset 14395
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