ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sess1 Unicode version

Theorem sess1 4059
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess1  R 
C_  S  S Se  R Se

Proof of Theorem sess1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . 6  R  C_  S  R  C_  S
21ssbrd 3796 . . . . 5  R  C_  S  R  S
32ss2rabdv 3015 . . . 4  R 
C_  S  {  |  R }  C_  {  |  S }
4 ssexg 3887 . . . . 5  {  |  R }  C_ 
{  |  S }  {  |  S }  _V  {  |  R }  _V
54ex 108 . . . 4  {  |  R }  C_  {  |  S }  {  |  S }  _V  {  |  R }  _V
63, 5syl 14 . . 3  R 
C_  S  {  |  S }  _V  {  |  R }  _V
76ralimdv 2382 . 2  R 
C_  S  {  |  S }  _V  {  |  R }  _V
8 df-se 4056 . 2  S Se  {  |  S }  _V
9 df-se 4056 . 2  R Se  {  |  R }  _V
107, 8, 93imtr4g 194 1  R 
C_  S  S Se  R Se
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1390  wral 2300   {crab 2304   _Vcvv 2551    C_ wss 2911   class class class wbr 3755   Se wse 4055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-br 3756  df-se 4056
This theorem is referenced by:  seeq1  4061
  Copyright terms: Public domain W3C validator