ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2nd0 Unicode version

Theorem 2nd0 5908
Description: The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
Assertion
Ref Expression
2nd0  |-  ( 2nd `  (/) )  =  (/)

Proof of Theorem 2nd0
StepHypRef Expression
1 0ex 3964 . . 3  |-  (/)  e.  _V
2 2ndvalg 5906 . . 3  |-  ( (/)  e.  _V  ->  ( 2nd `  (/) )  =  U. ran  { (/) } )
31, 2ax-mp 7 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
4 dmsn0 4893 . . . 4  |-  dom  { (/)
}  =  (/)
5 dm0rn0 4649 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
64, 5mpbi 143 . . 3  |-  ran  { (/)
}  =  (/)
76unieqi 3661 . 2  |-  U. ran  {
(/) }  =  U. (/)
8 uni0 3678 . 2  |-  U. (/)  =  (/)
93, 7, 83eqtri 2112 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438   _Vcvv 2619   (/)c0 3286   {csn 3444   U.cuni 3651   dom cdm 4436   ran crn 4437   ` cfv 5010   2ndc2nd 5902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-iota 4975  df-fun 5012  df-fv 5018  df-2nd 5904
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator