Theorem kmlem5 4769

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 2167	. . . 4 |- (w \ U.(x \ {w})) (_ w
2		sslin 2235	. . . 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 4768	. . . 4 |- ((w e. x /\ z =/= w) -> ((z \ U.(x \ {z})) i^i w) = (/))
5	4	sseq2d 2089	. . 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 2302	. 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 956   e. wcel 958   =/= wne 1585   \ cdif 2044   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503

This theorem is referenced by:  kmlem9 4773

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-sn 2412  df-pr 2413  df-uni 2504

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