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

Theorem 2ndvalg 6081
 Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
2ndvalg

Proof of Theorem 2ndvalg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 snexg 4140 . . 3
2 rnexg 4844 . . 3
3 uniexg 4394 . . 3
41, 2, 33syl 17 . 2
5 sneq 3567 . . . . 5
65rneqd 4808 . . . 4
76unieqd 3779 . . 3
8 df-2nd 6079 . . 3
97, 8fvmptg 5537 . 2
104, 9mpdan 418 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1332   wcel 2125  cvv 2709  csn 3556  cuni 3768   crn 4580  cfv 5163  c2nd 6077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fv 5171  df-2nd 6079 This theorem is referenced by:  2nd0  6083  op2nd  6085  elxp6  6107
 Copyright terms: Public domain W3C validator