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Theorem kmlem5 8523
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 3624 . . . 4  |-  ( w 
\  U. ( x  \  { w } ) )  C_  w
2 sslin 3717 . . . 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 8522 . . 3  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  w )  =  (/) )
53, 4syl5sseq 3545 . 2  |-  ( ( w  e.  x  /\  z  =/=  w )  -> 
( ( z  \  U. ( x  \  {
z } ) )  i^i  ( w  \  U. ( x  \  {
w } ) ) )  C_  (/) )
6 ss0b 3808 . 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 1374    =/= wne 2655    \ cdif 3466    i^i cin 3468    C_ wss 3469   (/)c0 3778   {csn 4020   U.cuni 4238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483  df-nul 3779  df-sn 4021  df-uni 4239
This theorem is referenced by:  kmlem9  8527
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