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Metamath Proof Explorer

Theorem qsdisj

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

Ref Expression
Hypotheses qsdisj.1 
|- ( ph -> R Er X )
qsdisj.2 
|- ( ph -> B e. ( A /. R ) )
qsdisj.3 
|- ( ph -> C e. ( A /. R ) )
Assertion qsdisj 
|- ( ph -> ( B = C \/ ( B i^i C ) = (/) ) )

Proof

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