Theorem elec 6145

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

Hypotheses

Ref	Expression
elec.1	|- A e. _V
elec.2	|- B e. _V

Assertion

Ref	Expression
elec	|- ( A e. [ B ] R <-> B R A )

Proof of Theorem elec

Step	Hyp	Ref	Expression
1		elec.1	. 2 |- A e. _V
2		elec.2	. 2 |- B e. _V
3		elecg 6144	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A e. [ B ] R <-> B R A ) )
4	1, 2, 3	mp2an 402	1 |- ( A e. [ B ] R <-> B R A )

Syntax hints:   <-> wb 98   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   [cec 6104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108

This theorem is referenced by:  ecid  6169