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Theorem elecALTV 37647
Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8748 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 6081 . 2 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2 df-ec 8707 . . 3 [𝐴]𝑅 = (𝑅 “ {𝐴})
32eleq2i 2819 . 2 (𝐵 ∈ [𝐴]𝑅𝐵 ∈ (𝑅 “ {𝐴}))
4 df-br 5142 . 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 2098  {csn 4623  cop 4629   class class class wbr 5141  cima 5672  [cec 8703
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707
This theorem is referenced by:  eldm4  37655  exan3  37676  exanres3  37678  ecin0  37734  dfcoss2  37796  eldm1cossres2  37844  eqvrelth  37994  eqvreldisj  37997  eqvrelqsel  37999  erimeq2  38061  disjlem19  38184
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