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

Theorem eqelsuc 4184
Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
Hypothesis
Ref Expression
eqelsuc.1  |-  A  e. 
_V
Assertion
Ref Expression
eqelsuc  |-  ( A  =  B  ->  A  e.  suc  B )

Proof of Theorem eqelsuc
StepHypRef Expression
1 eqelsuc.1 . . 3  |-  A  e. 
_V
21sucid 4182 . 2  |-  A  e. 
suc  A
3 suceq 4167 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
42, 3syl5eleq 2142 1  |-  ( A  =  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409   _Vcvv 2574   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-suc 4136
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator