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Theorem ispos 14409
 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 5731 . . . . . . 7
2 ispos.b . . . . . . 7
31, 2syl6eqr 2488 . . . . . 6
43eqeq2d 2449 . . . . 5
5 fveq2 5731 . . . . . . 7
6 ispos.l . . . . . . 7
75, 6syl6eqr 2488 . . . . . 6
87eqeq2d 2449 . . . . 5
94, 83anbi12d 1256 . . . 4
1092exbidv 1639 . . 3
11 df-poset 14408 . . 3
1210, 11elab4g 3088 . 2
13 fvex 5745 . . . . 5
142, 13eqeltri 2508 . . . 4
15 fvex 5745 . . . . 5
166, 15eqeltri 2508 . . . 4
17 raleq 2906 . . . . . 6
1817raleqbi1dv 2914 . . . . 5
1918raleqbi1dv 2914 . . . 4
20 breq 4217 . . . . . . 7
21 breq 4217 . . . . . . . . 9
22 breq 4217 . . . . . . . . 9
2321, 22anbi12d 693 . . . . . . . 8
2423imbi1d 310 . . . . . . 7
25 breq 4217 . . . . . . . . 9
2621, 25anbi12d 693 . . . . . . . 8
27 breq 4217 . . . . . . . 8
2826, 27imbi12d 313 . . . . . . 7
2920, 24, 283anbi123d 1255 . . . . . 6
3029ralbidv 2727 . . . . 5
31302ralbidv 2749 . . . 4
3214, 16, 19, 31ceqsex2v 2995 . . 3
3332anbi2i 677 . 2
3412, 33bitri 242 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  wral 2707  cvv 2958   class class class wbr 4215  cfv 5457  cbs 13474  cple 13541  cpo 14402 This theorem is referenced by:  ispos2  14410  posi  14412  0pos  14416  isposd  14417  isposi  14418  pospropd  14566  resspos  24192  xrstos  24206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-poset 14408
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