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Theorem ecid 8722
Description: A set is equal to its coset under the converse membership relation. (Note: the converse membership relation 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 3450 . . . 4 𝑦 ∈ V
2 ecid.1 . . . 4 𝐴 ∈ V
31, 2elec 8693 . . 3 (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦)
42, 1brcnv 5839 . . 3 (𝐴 E 𝑦𝑦 E 𝐴)
52epeli 5540 . . 3 (𝑦 E 𝐴𝑦𝐴)
63, 4, 53bitri 297 . 2 (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴)
76eqriv 2734 1 [𝐴] E = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3446   class class class wbr 5106   E cep 5537  ccnv 5633  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-eprel 5538  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8651
This theorem is referenced by:  qsid  8723  addcnsrec  11080  mulcnsrec  11081
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