Theorem elecALTV 39020

Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8742 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 6085	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		df-ec 8699	. . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
3	2	eleq2i 2852	. 2 ⊢ (𝐵 ∈ [𝐴]𝑅 ↔ 𝐵 ∈ (𝑅 “ {𝐴}))
4		df-br 5104	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
5	1, 3, 4	3bitr4g 317	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  {csn 4584  ⟨cop 4590   class class class wbr 5103   “ cima 5658  [cec 8695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8699

This theorem is used by:  eldm4  39030  exan3  39049  exanres3  39051  ecin0  39101  ecun  39142  ecxrn2  39157  dfsucmap3  39212  dfcoss2  39252  eldm1cossres2  39300  eqvrelth  39444  eqvreldisj  39447  eqvrelqsel  39449  erimeq2  39512  eldisjdmqsim  39566  disjlem19  39653