Theorem kmlem11 8561

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1

Assertion

Ref	Expression
kmlem11

Distinct variable groups:   ,,,   ,

Proof of Theorem kmlem11

Step	Hyp	Ref	Expression
1		kmlem9.1	. . . . . 6
2	1	unieqi 4258	. . . . 5
3		vex 3112	. . . . . . 7
4		difexg 4600	. . . . . . 7
5	3, 4	ax-mp 5	. . . . . 6
6	5	dfiun2 4364	. . . . 5
7	2, 6	eqtr4i 2489	. . . 4
8	7	ineq2i 3696	. . 3
9		iunin2 4394	. . 3
10	8, 9	eqtr4i 2489	. 2
11		undif2 3904	. . . . . 6
12		snssi 4174	. . . . . . 7
13		ssequn1 3673	. . . . . . 7
14	12, 13	sylib 196	. . . . . 6
15	11, 14	syl5req 2511	. . . . 5
16	15	iuneq1d 4355	. . . 4
17		iunxun 4412	. . . . . 6
18		vex 3112	. . . . . . . 8
19		difeq1 3614	. . . . . . . . . 10
20		sneq 4039	. . . . . . . . . . . . 13
21	20	difeq2d 3621	. . . . . . . . . . . 12
22	21	unieqd 4259	. . . . . . . . . . 11
23	22	difeq2d 3621	. . . . . . . . . 10
24	19, 23	eqtrd 2498	. . . . . . . . 9
25	24	ineq2d 3699	. . . . . . . 8
26	18, 25	iunxsn 4410	. . . . . . 7
27	26	uneq1i 3653	. . . . . 6
28	17, 27	eqtri 2486	. . . . 5
29		eldifsni 4156	. . . . . . . . . 10
30		incom 3690	. . . . . . . . . . . 12
31		kmlem4 8554	. . . . . . . . . . . 12
32	30, 31	syl5eq 2510	. . . . . . . . . . 11
33	32	ex 434	. . . . . . . . . 10
34	29, 33	syl5 32	. . . . . . . . 9
35	34	ralrimiv 2869	. . . . . . . 8
36		iuneq2 4347	. . . . . . . 8
37	35, 36	syl 16	. . . . . . 7
38		iun0 4386	. . . . . . 7
39	37, 38	syl6eq 2514	. . . . . 6
40	39	uneq2d 3657	. . . . 5
41	28, 40	syl5eq 2510	. . . 4
42	16, 41	eqtrd 2498	. . 3
43		un0 3810	. . . 4
44		indif 3739	. . . 4
45	43, 44	eqtri 2486	. . 3
46	42, 45	syl6eq 2514	. 2
47	10, 46	syl5eq 2510	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  \cdif 3472  u.cun 3473  i^icin 3474  C_wss 3475  c0 3784  {csn 4029  U.cuni 4249  U_ciun 4330

This theorem is referenced by:  kmlem12  8562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250  df-iun 4332