Theorem elec 6807

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 6806	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)

Syntax hints:   ↔ wb 105   ∈ wcel 2203  Vcvv 2812   class class class wbr 4108  [cec 6764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-cnv 4756  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-ec 6768

This theorem is referenced by:  ecid  6831