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Theorem ecid 8337
 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 3474 . . . 4 𝑦 ∈ V
2 ecid.1 . . . 4 𝐴 ∈ V
31, 2elec 8308 . . 3 (𝑦 ∈ [𝐴] E ↔ 𝐴 E 𝑦)
42, 1brcnv 5726 . . 3 (𝐴 E 𝑦𝑦 E 𝐴)
52epeli 5441 . . 3 (𝑦 E 𝐴𝑦𝐴)
63, 4, 53bitri 300 . 2 (𝑦 ∈ [𝐴] E ↔ 𝑦𝐴)
76eqriv 2818 1 [𝐴] E = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  Vcvv 3471   class class class wbr 5039   E cep 5437  ◡ccnv 5527  [cec 8262 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-eprel 5438  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ec 8266 This theorem is referenced by:  qsid  8338  addcnsrec  10542  mulcnsrec  10543
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