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Theorem elecg 7672
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
elecg ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))

Proof of Theorem elecg
StepHypRef Expression
1 elimasng 5410 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
21ancoms 468 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3 df-ec 7631 . . 3 [𝐵]𝑅 = (𝑅 “ {𝐵})
43eleq2i 2680 . 2 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ (𝑅 “ {𝐵}))
5 df-br 4584 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
62, 4, 53bitr4g 302 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wcel 1977  {csn 4125  cop 4131   class class class wbr 4583  cima 5041  [cec 7627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ec 7631
This theorem is referenced by:  elec  7673  relelec  7674  ecdmn0  7676  erth  7678  erdisj  7681  qsel  7713  orbsta  17569  sylow2alem1  17855  sylow2blem1  17858  sylow3lem3  17867  efgi2  17961  tgpconcompeqg  21725  xmetec  22049  blpnfctr  22051  xmetresbl  22052  xrsblre  22422
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