Theorem kmlem9 4753

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

Hypothesis

Ref	Expression
kmlem9.1	|- A = {u | E.t e. x u = (t \ U.(x \ {t}))}

Assertion

Ref	Expression
kmlem9	|- A.z e. A A.w e. A (z =/= w -> (z i^i w) = (/))

Distinct variable groups:   x,z,w,u,t   z,A,w

Proof of Theorem kmlem9

Step	Hyp	Ref	Expression
1		reeanv 1775	. . . 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 2208	. . . . . . . . . . 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	eqeq1d 1480	. . . . . . . . . 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 4749	. . . . . . . . . 10 |- ((h e. x /\ t =/= h) -> ((t \ U.(x \ {t})) i^i (h \ U.(x \ {h}))) = (/))
5	3, 4	syl5bir 210	. . . . . . . . 9 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> ((h e. x /\ t =/= h) -> (z i^i w) = (/)))
6	5	exp3a 375	. . . . . . . 8 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (h e. x -> (t =/= h -> (z i^i w) = (/))))
7		eqeq12 1484	. . . . . . . . . 10 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (z = w <-> (t \ U.(x \ {t})) = (h \ U.(x \ {h}))))
8		difeq1 2149	. . . . . . . . . . 11 |- (t = h -> (t \ U.(x \ {t})) = (h \ U.(x \ {t})))
9		sneq 2413	. . . . . . . . . . . . . 14 |- (t = h -> {t} = {h})
10	9	difeq2d 2155	. . . . . . . . . . . . 13 |- (t = h -> (x \ {t}) = (x \ {h}))
11	10	unieqd 2507	. . . . . . . . . . . 12 |- (t = h -> U.(x \ {t}) = U.(x \ {h}))
12	11	difeq2d 2155	. . . . . . . . . . 11 |- (t = h -> (h \ U.(x \ {t})) = (h \ U.(x \ {h})))
13	8, 12	eqtrd 1504	. . . . . . . . . 10 |- (t = h -> (t \ U.(x \ {t})) = (h \ U.(x \ {h})))
14	7, 13	syl5bir 210	. . . . . . . . 9 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (t = h -> z = w))
15	14	necon3d 1601	. . . . . . . 8 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (z =/= w -> t =/= h))
16	6, 15	syl5d 55	. . . . . . 7 |- ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (h e. x -> (z =/= w -> (z i^i w) = (/))))
17	16	com12 11	. . . . . 6 |- (h e. x -> ((z = (t \ U.(x \ {t})) /\ w = (h \ U.(x \ {h}))) -> (z =/= w -> (z i^i w) = (/))))
18	17	adantl 388	. . . . 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 1745	. . . 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 201	. . 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 1809	. . . 4 |- z e. V
22		eqeq1 1478	. . . . 5 |- (u = z -> (u = (t \ U.(x \ {t})) <-> z = (t \ U.(x \ {t}))))
23	22	rexbidv 1661	. . . 4 |- (u = z -> (E.t e. x u = (t \ U.(x \ {t})) <-> E.t e. x z = (t \ U.(x \ {t}))))
24		kmlem9.1	. . . 4 |- A = {u | E.t e. x u = (t \ U.(x \ {t}))}
25	21, 23, 24	elab2 1897	. . 3 |- (z e. A <-> E.t e. x z = (t \ U.(x \ {t})))
26		visset 1809	. . . . 5 |- w e. V
27		eqeq1 1478	. . . . . 6 |- (u = w -> (u = (t \ U.(x \ {t})) <-> w = (t \ U.(x \ {t}))))
28	27	rexbidv 1661	. . . . 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 1897	. . . 4 |- (w e. A <-> E.t e. x w = (t \ U.(x \ {t})))
30	13	eqeq2d 1483	. . . . 5 |- (t = h -> (w = (t \ U.(x \ {t})) <-> w = (h \ U.(x \ {h}))))
31	30	cbvrexv 1797	. . . 4 |- (E.t e. x w = (t \ U.(x \ {t})) <-> E.h e. x w = (h \ U.(x \ {h})))
32	29, 31	bitr 173	. . 3 |- (w e. A <-> E.h e. x w = (h \ U.(x \ {h})))
33	20, 25, 32	syl2anb 455	. 2 |- ((z e. A /\ w e. A) -> (z =/= w -> (z i^i w) = (/)))
34	33	rgen2a 1696	1 |- A.z e. A A.w e. A (z =/= w -> (z i^i w) = (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461   =/= wne 1582  A.wral 1642  E.wrex 1643   \ cdif 2040   i^i cin 2042  (/)c0 2276  {csn 2405  U.cuni 2498

This theorem is referenced by:  kmlem10 4754

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-rex 1647  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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