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Theorem kmlem5 7825
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 3337 . . . 4  |-  ( w 
\  U. ( x  \  { w } ) )  C_  w
2 sslin 3429 . . . 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 7824 . . 3  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  w )  =  (/) )
53, 4syl5sseq 3260 . 2  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  C_  (/) )
6 ss0b 3518 . 2  |-  ( ( ( z  \  U. ( x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) ) 
C_  (/)  <->  ( ( z 
\  U. ( x  \  { z } ) )  i^i  ( w 
\  U. ( x  \  { w } ) ) )  =  (/) )
75, 6sylib 188 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 358    = wceq 1633    e. wcel 1701    =/= wne 2479    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674   U.cuni 3864
This theorem is referenced by:  kmlem9  7829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-v 2824  df-dif 3189  df-in 3193  df-ss 3200  df-nul 3490  df-sn 3680  df-uni 3865
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