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

Theorem sseqtri 3032
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
Hypotheses
Ref Expression
sseqtr.1  |-  A  C_  B
sseqtr.2  |-  B  =  C
Assertion
Ref Expression
sseqtri  |-  A  C_  C

Proof of Theorem sseqtri
StepHypRef Expression
1 sseqtr.1 . 2  |-  A  C_  B
2 sseqtr.2 . . 3  |-  B  =  C
32sseq2i 3025 . 2  |-  ( A 
C_  B  <->  A  C_  C
)
41, 3mpbi 143 1  |-  A  C_  C
Colors of variables: wff set class
Syntax hints:    = 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:  sseqtr4i  3033  eqimssi  3054  abssi  3070  ssun2  3137  inssddif  3212  difdifdirss  3334  pwundifss  4048  unixpss  4479  0ima  4715
  Copyright terms: Public domain W3C validator