MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iinss1 Unicode version

Theorem iinss1 4073
Description: Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)
Assertion
Ref Expression
iinss1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iinss1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssralv 3375 . . 3  |-  ( A 
C_  B  ->  ( A. x  e.  B  y  e.  C  ->  A. x  e.  A  y  e.  C ) )
2 vex 2927 . . . 4  |-  y  e. 
_V
3 eliin 4066 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  B  C  <->  A. x  e.  B  y  e.  C ) )
42, 3ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  B  C 
<-> 
A. x  e.  B  y  e.  C )
5 eliin 4066 . . . 4  |-  ( y  e.  _V  ->  (
y  e.  |^|_ x  e.  A  C  <->  A. x  e.  A  y  e.  C ) )
62, 5ax-mp 8 . . 3  |-  ( y  e.  |^|_ x  e.  A  C 
<-> 
A. x  e.  A  y  e.  C )
71, 4, 63imtr4g 262 . 2  |-  ( A 
C_  B  ->  (
y  e.  |^|_ x  e.  B  C  ->  y  e.  |^|_ x  e.  A  C ) )
87ssrdv 3322 1  |-  ( A 
C_  B  ->  |^|_ x  e.  B  C  C_  |^|_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   A.wral 2674   _Vcvv 2924    C_ wss 3288   |^|_ciin 4062
This theorem is referenced by:  polcon3N  30411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-v 2926  df-in 3295  df-ss 3302  df-iin 4064
  Copyright terms: Public domain W3C validator