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Theorem eleccnvep 35419
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 5960 . . 3 Rel E
2 relelec 8323 . . 3 (Rel E → (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵))
31, 2ax-mp 5 . 2 (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵)
4 brcnvep 35407 . 2 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
53, 4syl5bb 284 1 (𝐴𝑉 → (𝐵 ∈ [𝐴] E ↔ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wcel 2105   class class class wbr 5057   E cep 5457  ccnv 5547  Rel wrel 5553  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by:  eccnvep  35420
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