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Theorem eleccnvep 34289
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 5613 . . 3 Rel E
2 relelec 7905 . . 3 (Rel E → (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵))
31, 2ax-mp 5 . 2 (𝐵 ∈ [𝐴] E ↔ 𝐴 E 𝐵)
4 brcnvep 34272 . 2 (𝐴𝑉 → (𝐴 E 𝐵𝐵𝐴))
53, 4syl5bb 272 1 (𝐴𝑉 → (𝐵 ∈ [𝐴] E ↔ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2103   class class class wbr 4760   E cep 5132  ccnv 5217  Rel wrel 5223  [cec 7860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-eprel 5133  df-xp 5224  df-rel 5225  df-cnv 5226  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-ec 7864
This theorem is referenced by:  eccnvep  34290
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