Theorem invdisj 4441

Description: If there is a function () such that ()=x for all e.(x), then the sets (x) for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)

Assertion

Ref	Expression
invdisj

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem invdisj

Step	Hyp	Ref	Expression
1		nfra2 2844	. . 3
2		df-ral 2812	. . . . 5
3		rsp 2823	. . . . . . . . 9
4		eqcom 2466	. . . . . . . . 9
5	3, 4	syl6ib 226	. . . . . . . 8
6	5	imim2i 14	. . . . . . 7
7	6	impd 431	. . . . . 6
8	7	alimi 1633	. . . . 5
9	2, 8	sylbi 195	. . . 4
10		mo2icl 3278	. . . 4
11	9, 10	syl 16	. . 3
12	1, 11	alrimi 1877	. 2
13		dfdisj2 4424	. 2
14	12, 13	sylibr 212	1

Syntax hints:  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  E*wmo 2283  A.wral 2807  Disj_wdisj 4422

This theorem is referenced by:  disjxwrd  12680  ackbijnn  13640  incexc2  13650  itg1addlem1  22099  musum  23467  lgsquadlem1  23629  lgsquadlem2  23630  disjabrex  27443  disjabrexf  27444  phisum  31159

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-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-ral 2812  df-rmo 2815  df-v 3111  df-disj 4423