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Theorem epelc 4264
Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
epelc.1 𝐵 ∈ V
Assertion
Ref Expression
epelc (𝐴 E 𝐵𝐴𝐵)

Proof of Theorem epelc
StepHypRef Expression
1 epelc.1 . 2 𝐵 ∈ V
2 epelg 4263 . 2 (𝐵 ∈ V → (𝐴 E 𝐵𝐴𝐵))
31, 2ax-mp 5 1 (𝐴 E 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 2135  Vcvv 2722   class class class wbr 3977   E cep 4260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-eprel 4262
This theorem is referenced by:  epel  4265  epini  4970  ecid  6556  ordiso2  6992
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