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Theorem eleccnvep 35646
Description: Elementhood in the converse epsilon coset of 𝐴 is elementhood in 𝐴. (Contributed by Peter Mazsa, 27-Jan-2019.)
Assertion
Ref Expression
eleccnvep (𝐴𝑉 → (𝐵 ∈ [𝐴] E ↔ 𝐵𝐴))

Proof of Theorem eleccnvep
StepHypRef Expression
1 relcnv 5954 . . 3 Rel E
2 relelec 8330 . . 3 (Rel E → (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵))
31, 2ax-mp 5 . 2 (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵)
4 brcnvep 35634 . 2 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
53, 4syl5bb 286 1 (𝐴𝑉 → (𝐵 ∈ [𝐴] E ↔ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2115   class class class wbr 5052   E cep 5451  ccnv 5541  Rel wrel 5547  [cec 8283
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 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-eprel 5452  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8287
This theorem is referenced by:  eccnvep  35647
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