MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inss Structured version   Unicode version

Theorem inss 3577
Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
Assertion
Ref Expression
inss  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )

Proof of Theorem inss
StepHypRef Expression
1 ssinss1 3576 . 2  |-  ( A 
C_  C  ->  ( A  i^i  B )  C_  C )
2 incom 3541 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
3 ssinss1 3576 . . 3  |-  ( B 
C_  C  ->  ( B  i^i  A )  C_  C )
42, 3syl5eqss 3398 . 2  |-  ( B 
C_  C  ->  ( A  i^i  B )  C_  C )
51, 4jaoi 379 1  |-  ( ( A  C_  C  \/  B  C_  C )  -> 
( A  i^i  B
)  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    i^i cin 3325    C_ wss 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-in 3333  df-ss 3340
This theorem is referenced by:  ppttop  18609  inindif  25895  pmatcoe1fsupp  30889
  Copyright terms: Public domain W3C validator