Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2ndnpr Unicode version

Theorem 2ndnpr 23261
Description: Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
2ndnpr  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 2nd `  A )  =  (/) )

Proof of Theorem 2ndnpr
StepHypRef Expression
1 2ndval 6141 . 2  |-  ( 2nd `  A )  =  U. ran  { A }
2 rnsnn0 5155 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
32biimpri 197 . . . . 5  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
43necon1bi 2502 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
54unieqd 3854 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
6 uni0 3870 . . 3  |-  U. (/)  =  (/)
75, 6syl6eq 2344 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
81, 7syl5eq 2340 1  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 2nd `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653   U.cuni 3843    X. cxp 4703   ran crn 4706   ` cfv 5271   2ndc2nd 6137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-2nd 6139
  Copyright terms: Public domain W3C validator