MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ecid Structured version   Visualization version   GIF version

Theorem ecid 8095
Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ecid.1 𝐴 ∈ V
Assertion
Ref Expression
ecid [𝐴] E = 𝐴

Proof of Theorem ecid
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3401 . . . 4 𝑦 ∈ V
2 ecid.1 . . . 4 𝐴 ∈ V
31, 2elec 8068 . . 3 (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦)
42, 1brcnv 5550 . . 3 (𝐴 E 𝑦𝑦 E 𝐴)
52epeli 5268 . . 3 (𝑦 E 𝐴𝑦𝐴)
63, 4, 53bitri 289 . 2 (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴)
76eqriv 2775 1 [𝐴] E = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  Vcvv 3398   class class class wbr 4886   E cep 5265  ccnv 5354  [cec 8024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-eprel 5266  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ec 8028
This theorem is referenced by:  qsid  8096  addcnsrec  10300  mulcnsrec  10301
  Copyright terms: Public domain W3C validator