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Theorem ispos 15427
 Description: The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
Hypotheses
Ref Expression
ispos.b
ispos.l
Assertion
Ref Expression
ispos
Distinct variable groups:   ,,,   , ,,
Allowed substitution hints:   (,,)

Proof of Theorem ispos
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5864 . . . . . . 7
2 ispos.b . . . . . . 7
31, 2syl6eqr 2526 . . . . . 6
43eqeq2d 2481 . . . . 5
5 fveq2 5864 . . . . . . 7
6 ispos.l . . . . . . 7
75, 6syl6eqr 2526 . . . . . 6
87eqeq2d 2481 . . . . 5
94, 83anbi12d 1300 . . . 4
1092exbidv 1692 . . 3
11 df-poset 15426 . . 3
1210, 11elab4g 3254 . 2
13 fvex 5874 . . . . 5
142, 13eqeltri 2551 . . . 4
15 fvex 5874 . . . . 5
166, 15eqeltri 2551 . . . 4
17 raleq 3058 . . . . . 6
1817raleqbi1dv 3066 . . . . 5
1918raleqbi1dv 3066 . . . 4
20 breq 4449 . . . . . . 7
21 breq 4449 . . . . . . . . 9
22 breq 4449 . . . . . . . . 9
2321, 22anbi12d 710 . . . . . . . 8
2423imbi1d 317 . . . . . . 7
25 breq 4449 . . . . . . . . 9
2621, 25anbi12d 710 . . . . . . . 8
27 breq 4449 . . . . . . . 8
2826, 27imbi12d 320 . . . . . . 7
2920, 24, 283anbi123d 1299 . . . . . 6
3029ralbidv 2903 . . . . 5
31302ralbidv 2908 . . . 4
3214, 16, 19, 31ceqsex2v 3152 . . 3
3332anbi2i 694 . 2
3412, 33bitri 249 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1379  wex 1596   wcel 1767  wral 2814  cvv 3113   class class class wbr 4447  cfv 5586  cbs 14483  cple 14555  cpo 15420 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-poset 15426 This theorem is referenced by:  ispos2  15428  posi  15430  0pos  15434  isposd  15435  isposi  15436  pospropd  15614  resspos  27306  xrstos  27326  xrge0omnd  27360
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