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Theorem relelec 7830
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 3243 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ V)
2 ecexr 7792 . . . 4 (𝐴 ∈ [𝐵]𝑅𝐵 ∈ V)
31, 2jca 553 . . 3 (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
43adantl 481 . 2 ((Rel 𝑅𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 brrelex12 5189 . . 3 ((Rel 𝑅𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
65ancomd 466 . 2 ((Rel 𝑅𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 elecg 7828 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
84, 6, 7pm5.21nd 961 1 (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  Vcvv 3231   class class class wbr 4685  Rel wrel 5148  [cec 7785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ec 7789
This theorem is referenced by:  eqgid  17693  tgptsmscls  22000  pstmfval  30067  ismntop  30198  topfneec  32475  releleccnv  34162  elecres  34183  eleccnvep  34187  inecmo  34260  elecxrn  34288  elec1cnvxrn2  34295  eleccossin  34373
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