Theorem kmlem5 8976

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

Step	Hyp	Ref	Expression
1		difss 3737	. . . 4 |- ( w \ U. ( x \ { w } ) ) C_ w
2		sslin 3839	. . . 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 ) )
3	1, 2	ax-mp 5	. . 3 |- ( ( z \ U. ( x \ { z } ) ) i^i ( w \ U. ( x \ { w } ) ) ) C_ ( ( z \ U. ( x \ { z } ) ) i^i w )
4		kmlem4 8975	. . 3 |- ( ( w e. x /\ z =/= w ) -> ( ( z \ U. ( x \ { z } ) ) i^i w ) = (/) )
5	3, 4	syl5sseq 3653	. 2 |- ( ( w e. x /\ z =/= w ) -> ( ( z \ U. ( x \ { z } ) ) i^i ( w \ U. ( x \ { w } ) ) ) C_ (/) )
6		ss0b 3973	. 2 |- ( ( ( z \ U. ( x \ { z } ) ) i^i ( w \ U. ( x \ { w } ) ) ) C_ (/) <-> ( ( z \ U. ( x \ { z } ) ) i^i ( w \ U. ( x \ { w } ) ) ) = (/) )
7	5, 6	sylib 208	1 |- ( ( w e. x /\ z =/= w ) -> ( ( z \ U. ( x \ { z } ) ) i^i ( w \ U. ( x \ { w } ) ) ) = (/) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437

This theorem is referenced by:  kmlem9  8980