Theorem elecALTV 38255

Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8715 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 6060	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		df-ec 8673	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
3	2	eleq2i 2820	. 2 ⊢ (𝐵 ∈ [𝐴]𝑅 ↔ 𝐵 ∈ (𝑅 “ {𝐴}))
4		df-br 5108	. 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 206   ∧ wa 395   ∈ wcel 2109  {csn 4589  ⟨cop 4595   class class class wbr 5107   “ cima 5641  [cec 8669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673

This theorem is referenced by:  eldm4  38263  exan3  38282  exanres3  38284  ecin0  38334  dfcoss2  38404  eldm1cossres2  38452  eqvrelth  38602  eqvreldisj  38605  eqvrelqsel  38607  erimeq2  38670  disjlem19  38793