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Theorem relelec 6476
Description: Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
relelec (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Proof of Theorem relelec
StepHypRef Expression
1 elex 2700 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ V)
2 ecexr 6441 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
31, 2jca 304 . . 3 (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43adantl 275 . 2 ((Rel 𝑅𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 brrelex12 4584 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
65ancomd 265 . 2 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 elecg 6474 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
84, 6, 7pm5.21nd 902 1 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1481  Vcvv 2689   class class class wbr 3936  Rel wrel 4551  [cec 6434
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-ec 6438
This theorem is referenced by: (None)
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