Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erprt Structured version   Visualization version   Unicode version
Theorem erprt 34158
Description: The quotient set of an equivalence relation is a partition. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
erprt |- ( .~ Er X -> Prt ( A /. .~ ) )
Proof of Theorem erprt
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 473 . . . 4 |- ( ( .~ Er X /\ ( x e. ( A /. .~ ) /\ y e. ( A /. .~ ) ) ) -> .~ Er X )
2 simprl 794 . . . 4 |- ( ( .~ Er X /\ ( x e. ( A /. .~ ) /\ y e. ( A /. .~ ) ) ) -> x e. ( A /. .~ ) )
3 simprr 796 . . . 4 |- ( ( .~ Er X /\ ( x e. ( A /. .~ ) /\ y e. ( A /. .~ ) ) ) -> y e. ( A /. .~ ) )
41, 2, 3qsdisj 7824 . . 3 |- ( ( .~ Er X /\ ( x e. ( A /. .~ ) /\ y e. ( A /. .~ ) ) ) -> ( x = y \/ ( x i^i y ) = (/) ) )
54ralrimivva 2971 . 2 |- ( .~ Er X -> A. x e. ( A /. .~ ) A. y e. ( A /. .~ ) ( x = y \/ ( x i^i y ) = (/) ) )
6 df-prt 34157 . 2 |- ( Prt ( A /. .~ ) <-> A. x e. ( A /. .~ ) A. y e. ( A /. .~ ) ( x = y \/ ( x i^i y ) = (/) ) )
75, 6sylibr 224 1 |- ( .~ Er X -> Prt ( A /. .~ ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   (/)c0 3915   Er wer 7739   /.cqs 7741   Prt wprt 34156
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  df-prt 34157
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator