Theorem elec 6691

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 6690	. 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 2180  Vcvv 2779   class class class wbr 4062  [cec 6648

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-ec 6652

This theorem is referenced by:  ecid  6715