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Theorem 2ndnpr 23936
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 6292 . 2  |-  ( 2nd `  A )  =  U. ran  { A }
2 rnsnn0 5277 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
32biimpri 198 . . . . 5  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
43necon1bi 2594 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
54unieqd 3969 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
6 uni0 3985 . . 3  |-  U. (/)  =  (/)
75, 6syl6eq 2436 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
81, 7syl5eq 2432 1  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( 2nd `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572   {csn 3758   U.cuni 3958    X. cxp 4817   ran crn 4820   ` cfv 5395   2ndc2nd 6288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403  df-2nd 6290
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