Theorem disji2 4439

Description: Property of a disjoint collection: if ()= and ()=, and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disji.1
disji.2

Assertion

Ref	Expression
disji2

Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem disji2

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ne 2654	. . 3
2		disjors 4438	. . . . . 6
3		eqeq1 2461	. . . . . . . 8
4		nfcv 2619	. . . . . . . . . . 11
5		nfcv 2619	. . . . . . . . . . 11
6		disji.1	. . . . . . . . . . 11
7	4, 5, 6	csbhypf 3453	. . . . . . . . . 10
8	7	ineq1d 3698	. . . . . . . . 9
9	8	eqeq1d 2459	. . . . . . . 8
10	3, 9	orbi12d 709	. . . . . . 7
11		eqeq2 2472	. . . . . . . 8
12		nfcv 2619	. . . . . . . . . . 11
13		nfcv 2619	. . . . . . . . . . 11
14		disji.2	. . . . . . . . . . 11
15	12, 13, 14	csbhypf 3453	. . . . . . . . . 10
16	15	ineq2d 3699	. . . . . . . . 9
17	16	eqeq1d 2459	. . . . . . . 8
18	11, 17	orbi12d 709	. . . . . . 7
19	10, 18	rspc2v 3219	. . . . . 6
20	2, 19	syl5bi 217	. . . . 5
21	20	impcom 430	. . . 4
22	21	ord 377	. . 3
23	1, 22	syl5bi 217	. 2
24	23	3impia 1193	1

Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  [_csb 3434  i^icin 3474  c0 3784  Disj_wdisj 4422

This theorem is referenced by:  disji  4440  disjxiun  4449  voliunlem1  21960

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

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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-in 3482  df-nul 3785  df-disj 4423