Theorem elec 8683

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

Hypotheses

Ref	Expression
elec.1	⊢ 𝐴 ∈ V
elec.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
elec	⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)

Proof of Theorem elec

Step	Hyp	Ref	Expression
1		elec.1	. 2 ⊢ 𝐴 ∈ V
2		elec.2	. 2 ⊢ 𝐵 ∈ V
3		elecg 8681	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3430   class class class wbr 5086  [cec 8634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8638

This theorem is referenced by:  ecid  8720  sylow2alem2  19584  sylow2a  19585  sylow2blem1  19586  efgval2  19690  efgrelexlemb  19716  efgcpbllemb  19721  frgpnabllem1  19839  tgpconncomp  24088  qustgphaus  24098  vitalilem2  25586  vitalilem3  25587  isbndx  38117  prtlem10  39325  prtlem19  39338  prter3  39342