ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sseq2 Unicode version

Theorem sseq2 3022
Description: Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
Assertion
Ref Expression
sseq2  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )

Proof of Theorem sseq2
StepHypRef Expression
1 sstr2 3007 . . . 4  |-  ( C 
C_  A  ->  ( A  C_  B  ->  C  C_  B ) )
21com12 30 . . 3  |-  ( A 
C_  B  ->  ( C  C_  A  ->  C  C_  B ) )
3 sstr2 3007 . . . 4  |-  ( C 
C_  B  ->  ( B  C_  A  ->  C  C_  A ) )
43com12 30 . . 3  |-  ( B 
C_  A  ->  ( C  C_  B  ->  C  C_  A ) )
52, 4anim12i 331 . 2  |-  ( ( A  C_  B  /\  B  C_  A )  -> 
( ( C  C_  A  ->  C  C_  B
)  /\  ( C  C_  B  ->  C  C_  A
) ) )
6 eqss 3015 . 2  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
7 dfbi2 380 . 2  |-  ( ( C  C_  A  <->  C  C_  B
)  <->  ( ( C 
C_  A  ->  C  C_  B )  /\  ( C  C_  B  ->  C  C_  A ) ) )
85, 6, 73imtr4i 199 1  |-  ( A  =  B  ->  ( C  C_  A  <->  C  C_  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987
This theorem is referenced by:  sseq12  3023  sseq2i  3025  sseq2d  3028  syl5sseq  3048  nssne1  3056  sseq0  3292  un00  3297  pweq  3393  ssintab  3661  ssintub  3662  intmin  3664  treq  3889  ssexg  3925  frforeq3  4110  frirrg  4113  iunpw  4237  ordtri2orexmid  4274  ontr2exmid  4276  onsucsssucexmid  4278  ordtri2or2exmid  4322  fununi  4998  funcnvuni  4999  feq3  5063  ssimaexg  5267  nnawordex  6167  ereq1  6179  xpiderm  6243  domeng  6299  ssfiexmid  6411  bdssexg  10853  bj-nntrans  10904  bj-omtrans  10909
  Copyright terms: Public domain W3C validator