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Theorem relelec 6346
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 2631 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ V)
2 ecexr 6311 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
31, 2jca 301 . . 3 (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43adantl 272 . 2 ((Rel 𝑅𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 brrelex12 4489 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
65ancomd 264 . 2 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 elecg 6344 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
84, 6, 7pm5.21nd 864 1 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1439  Vcvv 2620   class class class wbr 3851  Rel wrel 4457  [cec 6304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-ec 6308
This theorem is referenced by: (None)
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