MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kmlem5 Structured version   Unicode version

Theorem kmlem5 8034
Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
kmlem5  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  =  (/) )
Distinct variable group:    x, w, z

Proof of Theorem kmlem5
StepHypRef Expression
1 difss 3474 . . . 4  |-  ( w 
\  U. ( x  \  { w } ) )  C_  w
2 sslin 3567 . . . 4  |-  ( ( w  \  U. (
x  \  { w } ) )  C_  w  ->  ( ( z 
\  U. ( x  \  { z } ) )  i^i  ( w 
\  U. ( x  \  { w } ) ) )  C_  (
( z  \  U. ( x  \  { z } ) )  i^i  w ) )
31, 2ax-mp 8 . . 3  |-  ( ( z  \  U. (
x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) ) 
C_  ( ( z 
\  U. ( x  \  { z } ) )  i^i  w )
4 kmlem4 8033 . . 3  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  w )  =  (/) )
53, 4syl5sseq 3396 . 2  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  C_  (/) )
6 ss0b 3657 . 2  |-  ( ( ( z  \  U. ( x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) ) 
C_  (/)  <->  ( ( z 
\  U. ( x  \  { z } ) )  i^i  ( w 
\  U. ( x  \  { w } ) ) )  =  (/) )
75, 6sylib 189 1  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   U.cuni 4015
This theorem is referenced by:  kmlem9  8038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-uni 4016
  Copyright terms: Public domain W3C validator