Theorem kmlem5 4749

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4.

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 2163	. . . 4 |- (w \ U.(x \ {w})) (_ w
2		sslin 2231	. . . 4 |- ((w \ U.(x \ {w})) (_ w -> ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ ((z \ U.(x \ {z})) i^i w))
3	1, 2	ax-mp 7	. . 3 |- ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ ((z \ U.(x \ {z})) i^i w)
4		kmlem4 4748	. . . 4 |- ((w e. x /\ z =/= w) -> ((z \ U.(x \ {z})) i^i w) = (/))
5	4	sseq2d 2085	. . 3 |- ((w e. x /\ z =/= w) -> (((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ ((z \ U.(x \ {z})) i^i w) <-> ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ (/)))
6	3, 5	mpbii 193	. 2 |- ((w e. x /\ z =/= w) -> ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ (/))
7		ss0b 2298	. 2 |- (((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) (_ (/) <-> ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) = (/))
8	6, 7	sylib 198	1 |- ((w e. x /\ z =/= w) -> ((z \ U.(x \ {z})) i^i (w \ U.(x \ {w}))) = (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582   \ cdif 2040   i^i cin 2042   (_ wss 2043  (/)c0 2276  {csn 2405  U.cuni 2498

This theorem is referenced by:  kmlem9 4753

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-uni 2499

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