Theorem kmlem8 3587

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

Hypothesis

Ref	Expression
kmlem8.1	|- A = {u | E.t e. x u = (t \ U.(x \ {t}))}

Assertion

Ref	Expression
kmlem8	|- A.z e. A A.w e. A (-. z = w -> (z i^i w) = (/))

Distinct variable group(s):   x,z,w,u,t   z,A,w

Proof of Theorem kmlem8

Step	Hyp	Ref	Expression
1		reeanv 1316	. . . 4 |- (E.t e. x E.h e. x (z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) <-> (E.t e. x z = (t \ U.(x \ {t})) /\ E.h e. x w = (h \ U.(x \ {h}))))
2		ineq12 1640	. . . . . . . . . . 11 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (z i^i w) = ((t \ U.(x \ {t})) i^i (h \ U.(x \ {h}))))
3	2	cleq1d 1109	. . . . . . . . . 10 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> ((z i^i w) = (/) <-> ((t \ U.(x \ {t})) i^i (h \ U.(x \ {h}))) = (/)))
4		kmlem5 3584	. . . . . . . . . 10 |- ((h e. x /\ -. t = h) -> ((t \ U.(x \ {t})) i^i (h \ U.(x \ {h}))) = (/))
5	3, 4	syl5bir 184	. . . . . . . . 9 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> ((h e. x /\ -. t = h) -> (z i^i w) = (/)))
6	5	exp3a 292	. . . . . . . 8 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (h e. x -> (-. t = h -> (z i^i w) = (/))))
7		cleq12 1113	. . . . . . . . . 10 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (z = w <-> (t \ U.(x \ {t})) = (h \ U.(x \ {h}))))
8		difeq1 1582	. . . . . . . . . . 11 |- (t = h -> (t \ U.(x \ {t})) = (h \ U.(x \ {t})))
9		sneq 1816	. . . . . . . . . . . . . 14 |- (t = h -> {t} = {h})
10	9	difeq2d 1588	. . . . . . . . . . . . 13 |- (t = h -> (x \ {t}) = (x \ {h}))
11	10	unieqd 1929	. . . . . . . . . . . 12 |- (t = h -> U.(x \ {t}) = U.(x \ {h}))
12	11	difeq2d 1588	. . . . . . . . . . 11 |- (t = h -> (h \ U.(x \ {t})) = (h \ U.(x \ {h})))
13	8, 12	eqtrd 1128	. . . . . . . . . 10 |- (t = h -> (t \ U.(x \ {t})) = (h \ U.(x \ {h})))
14	7, 13	syl5bir 184	. . . . . . . . 9 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (t = h -> z = w))
15	14	con3d 87	. . . . . . . 8 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (-. z = w -> -. t = h))
16	6, 15	syl5d 53	. . . . . . 7 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (h e. x -> (-. z = w -> (z i^i w) = (/))))
17	16	com12 13	. . . . . 6 |- (h e. x -> ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (-. z = w -> (z i^i w) = (/))))
18	17	adantl 305	. . . . 5 |- ((t e. x /\ h e. x) -> ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (-. z = w -> (z i^i w) = (/))))
19	18	r19.23aivv 1287	. . . 4 |- (E.t e. x E.h e. x (z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (-. z = w -> (z i^i w) = (/)))
20	1, 19	sylbir 176	. . 3 |- ((E.t e. x z = (t \ U.(x \ {t})) /\ E.h e. x w = (h \ U.(x \ {h}))) -> (-. z = w -> (z i^i w) = (/)))
21		visset 1350	. . . 4 |- z e. V
22		cleq1 1107	. . . . 5 |- (u = z -> (u = (t \ U.(x \ {t})) <-> z = (t \ U.(x \ {t}))))
23	22	birexdv 1220	. . . 4 |- (u = z -> (E.t e. x u = (t \ U.(x \ {t})) <-> E.t e. x z = (t \ U.(x \ {t}))))
24		kmlem8.1	. . . 4 |- A = {u | E.t e. x u = (t \ U.(x \ {t}))}
25	21, 23, 24	elab2 1419	. . 3 |- (z e. A <-> E.t e. x z = (t \ U.(x \ {t})))
26		visset 1350	. . . . 5 |- w e. V
27		cleq1 1107	. . . . . 6 |- (u = w -> (u = (t \ U.(x \ {t})) <-> w = (t \ U.(x \ {t}))))
28	27	birexdv 1220	. . . . 5 |- (u = w -> (E.t e. x u = (t \ U.(x \ {t})) <-> E.t e. x w = (t \ U.(x \ {t}))))
29	26, 28, 24	elab2 1419	. . . 4 |- (w e. A <-> E.t e. x w = (t \ U.(x \ {t})))
30	13	cleq2d 1112	. . . . 5 |- (t = h -> (w = (t \ U.(x \ {t})) <-> w = (h \ U.(x \ {h}))))
31	30	cbvrexv 1334	. . . 4 |- (E.t e. x w = (t \ U.(x \ {t})) <-> E.h e. x w = (h \ U.(x \ {h})))
32	29, 31	bitr 151	. . 3 |- (w e. A <-> E.h e. x w = (h \ U.(x \ {h})))
33	20, 25, 32	syl2anb 350	. 2 |- ((z e. A /\ w e. A) -> (-. z = w -> (z i^i w) = (/)))
34	33	rgen2 1248	1 |- A.z e. A A.w e. A (-. z = w -> (z i^i w) = (/))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   \ cdif 1484   i^i cin 1486  (/)c0 1707  {csn 1808  U.cuni 1919

This theorem is referenced by:  kmlem9 3588

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-uni 1920

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