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Theorem elecALTV 37794
Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8766 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 6087 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2 df-ec 8725 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
32eleq2i 2817 . 2 (𝐵 ∈ [𝐴]𝑅𝐵 ∈ (𝑅 “ {𝐴}))
4 df-br 5144 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
51, 3, 43bitr4g 313 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  {csn 4624  cop 4630   class class class wbr 5143  cima 5675  [cec 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725
This theorem is referenced by:  eldm4  37802  exan3  37822  exanres3  37824  ecin0  37880  dfcoss2  37941  eldm1cossres2  37989  eqvrelth  38139  eqvreldisj  38142  eqvrelqsel  38144  erimeq2  38206  disjlem19  38329
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