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Theorem elecg 8051
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 5733 . . 3 ((𝐵𝑊𝐴𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
21ancoms 452 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
3 df-ec 8012 . . 3 [𝐵]𝑅 = (𝑅 “ {𝐵})
43eleq2i 2899 . 2 (𝐴 ∈ [𝐵]𝑅𝐴 ∈ (𝑅 “ {𝐵}))
5 df-br 4875 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
62, 4, 53bitr4g 306 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ∈ [𝐵]𝑅𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2166  {csn 4398  cop 4404   class class class wbr 4874  cima 5346  [cec 8008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-ec 8012
This theorem is referenced by:  elec  8052  relelec  8053  ecdmn0  8055  erth  8057  erdisj  8060  qsel  8092  orbsta  18097  sylow2alem1  18384  sylow2blem1  18387  sylow3lem3  18396  efgi2  18490  tgpconncompeqg  22286  xmetec  22610  blpnfctr  22612  xmetresbl  22613  xrsblre  22985  ecin0  34666  eqvrelth  34902
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