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Theorem kmlem5 8435
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 3592 . . . 4  |-  ( w 
\  U. ( x  \  { w } ) )  C_  w
2 sslin 3685 . . . 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 5 . . 3  |-  ( ( z  \  U. (
x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) ) 
C_  ( ( z 
\  U. ( x  \  { z } ) )  i^i  w )
4 kmlem4 8434 . . 3  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  w )  =  (/) )
53, 4syl5sseq 3513 . 2  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  C_  (/) )
6 ss0b 3776 . 2  |-  ( ( ( z  \  U. ( x  \  { z } ) )  i^i  ( w  \  U. ( x  \  { w } ) ) ) 
C_  (/)  <->  ( ( z 
\  U. ( x  \  { z } ) )  i^i  ( w 
\  U. ( x  \  { w } ) ) )  =  (/) )
75, 6sylib 196 1  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    =/= wne 2648    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3746   {csn 3986   U.cuni 4200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747  df-sn 3987  df-uni 4201
This theorem is referenced by:  kmlem9  8439
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