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Theorem elecALTV 37732
Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8761 with this original form of Suppes. Peter Mazsa). (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
elecALTV ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))

Proof of Theorem elecALTV
StepHypRef Expression
1 elimasng 6086 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2 df-ec 8720 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
32eleq2i 2821 . 2 (𝐵 ∈ [𝐴]𝑅𝐵 ∈ (𝑅 “ {𝐴}))
4 df-br 5143 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
51, 3, 43bitr4g 314 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  {csn 4624  cop 4630   class class class wbr 5142  cima 5675  [cec 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8720
This theorem is referenced by:  eldm4  37740  exan3  37760  exanres3  37762  ecin0  37818  dfcoss2  37879  eldm1cossres2  37927  eqvrelth  38077  eqvreldisj  38080  eqvrelqsel  38082  erimeq2  38144  disjlem19  38267
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