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Theorem isprs 14379
 Description: Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypotheses
Ref Expression
isprs.b
isprs.l
Assertion
Ref Expression
isprs
Distinct variable groups:   ,,,   ,,,   , ,,

Proof of Theorem isprs
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . 5
2 dfsbcq 3155 . . . . 5
31, 2syl 16 . . . 4
4 fveq2 5720 . . . . . 6
5 dfsbcq 3155 . . . . . 6
64, 5syl 16 . . . . 5
76sbcbidv 3207 . . . 4
83, 7bitrd 245 . . 3
9 fvex 5734 . . . 4
10 fvex 5734 . . . 4
11 isprs.b . . . . . . 7
12 eqtr3 2454 . . . . . . 7
1311, 12mpan2 653 . . . . . 6
14 raleq 2896 . . . . . . . 8
1514raleqbi1dv 2904 . . . . . . 7
1615raleqbi1dv 2904 . . . . . 6
1713, 16syl 16 . . . . 5
18 isprs.l . . . . . . 7
19 eqtr3 2454 . . . . . . 7
2018, 19mpan2 653 . . . . . 6
21 breq 4206 . . . . . . . . 9
22 breq 4206 . . . . . . . . . . 11
23 breq 4206 . . . . . . . . . . 11
2422, 23anbi12d 692 . . . . . . . . . 10
25 breq 4206 . . . . . . . . . 10
2624, 25imbi12d 312 . . . . . . . . 9
2721, 26anbi12d 692 . . . . . . . 8
2827ralbidv 2717 . . . . . . 7
29282ralbidv 2739 . . . . . 6
3020, 29syl 16 . . . . 5
3117, 30sylan9bb 681 . . . 4
329, 10, 31sbc2ie 3220 . . 3
338, 32syl6bb 253 . 2
34 df-preset 14377 . 2
3533, 34elab4g 3078 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  cvv 2948  wsbc 3153   class class class wbr 4204  cfv 5446  cbs 13461  cple 13528   cpreset 14375 This theorem is referenced by:  prslem  14380  ispos2  14397 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
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