Theorem qsdisj 7824

Description: Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)

Hypotheses

Ref	Expression
qsdisj.1	|- ( ph -> R Er X )
qsdisj.2	|- ( ph -> B e. ( A /. R ) )
qsdisj.3	|- ( ph -> C e. ( A /. R ) )

Assertion

Ref	Expression
qsdisj	|- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) )

Proof of Theorem qsdisj

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

Step	Hyp	Ref	Expression
1		qsdisj.2	. 2 |- ( ph -> B e. ( A /. R ) )
2		eqid 2622	. . 3 |- ( A /. R ) = ( A /. R )
3		eqeq1 2626	. . . 4 |- ( [ x ] R = B -> ( [ x ] R = C <-> B = C ) )
4		ineq1 3807	. . . . 5 |- ( [ x ] R = B -> ( [ x ] R i^i C ) = ( B i^i C ) )
5	4	eqeq1d 2624	. . . 4 |- ( [ x ] R = B -> ( ( [ x ] R i^i C ) = (/) <-> ( B i^i C ) = (/) ) )
6	3, 5	orbi12d 746	. . 3 |- ( [ x ] R = B -> ( ( [ x ] R = C \/ ( [ x ] R i^i C ) = (/) ) <-> ( B = C \/ ( B i^i C ) = (/) ) ) )
7		qsdisj.3	. . . 4 |- ( ph -> C e. ( A /. R ) )
8		eqeq2 2633	. . . . . 6 |- ( [ y ] R = C -> ( [ x ] R = [ y ] R <-> [ x ] R = C ) )
9		ineq2 3808	. . . . . . 7 |- ( [ y ] R = C -> ( [ x ] R i^i [ y ] R ) = ( [ x ] R i^i C ) )
10	9	eqeq1d 2624	. . . . . 6 |- ( [ y ] R = C -> ( ( [ x ] R i^i [ y ] R ) = (/) <-> ( [ x ] R i^i C ) = (/) ) )
11	8, 10	orbi12d 746	. . . . 5 |- ( [ y ] R = C -> ( ( [ x ] R = [ y ] R \/ ( [ x ] R i^i [ y ] R ) = (/) ) <-> ( [ x ] R = C \/ ( [ x ] R i^i C ) = (/) ) ) )
12		qsdisj.1	. . . . . . 7 |- ( ph -> R Er X )
13	12	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. A ) /\ y e. A ) -> R Er X )
14		erdisj 7794	. . . . . 6 |- ( R Er X -> ( [ x ] R = [ y ] R \/ ( [ x ] R i^i [ y ] R ) = (/) ) )
15	13, 14	syl 17	. . . . 5 |- ( ( ( ph /\ x e. A ) /\ y e. A ) -> ( [ x ] R = [ y ] R \/ ( [ x ] R i^i [ y ] R ) = (/) ) )
16	2, 11, 15	ectocld 7814	. . . 4 |- ( ( ( ph /\ x e. A ) /\ C e. ( A /. R ) ) -> ( [ x ] R = C \/ ( [ x ] R i^i C ) = (/) ) )
17	7, 16	mpidan 704	. . 3 |- ( ( ph /\ x e. A ) -> ( [ x ] R = C \/ ( [ x ] R i^i C ) = (/) ) )
18	2, 6, 17	ectocld 7814	. 2 |- ( ( ph /\ B e. ( A /. R ) ) -> ( B = C \/ ( B i^i C ) = (/) ) )
19	1, 18	mpdan 702	1 |- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   (/)c0 3915   Er wer 7739   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744  df-qs 7748

This theorem is referenced by:  qsdisj2  7825  uniinqs  7827  cldsubg  21914  erprt  34158