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Theorem elecALTV 35408
Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8321 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 5948 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2 df-ec 8280 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
32eleq2i 2901 . 2 (𝐵 ∈ [𝐴]𝑅𝐵 ∈ (𝑅 “ {𝐴}))
4 df-br 5058 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
51, 3, 43bitr4g 315 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ [𝐴]𝑅𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  {csn 4557  cop 4563   class class class wbr 5057  cima 5551  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by:  eldm4  35412  exan3  35432  exanres3  35434  ecin0  35487  dfcoss2  35541  eldm1cossres2  35581  eqvrelth  35726  eqvreldisj  35729  eqvrelqsel  35731  erim2  35791
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