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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵))

Proof of Theorem elecALTV

Step	Hyp	Ref	Expression
1		elimasng 6086	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		df-ec 8720	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
3	2	eleq2i 2821	. 2 ⊢ (𝐵 ∈ [𝐴]𝑅 ↔ 𝐵 ∈ (𝑅 “ {𝐴}))
4		df-br 5143	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
5	1, 3, 4	3bitr4g 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵))

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