Theorem elec 4279

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82.

Hypotheses

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

Assertion

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

Proof of Theorem elec

Step	Hyp	Ref	Expression
1		elec.1	. 2 |- A e. V
2		breq2 2623	. 2 |- (x = A -> (BRx <-> BRA))
3		elec.2	. . 3 |- B e. V
4	3	dfec2 4264	. 2 |- [B]R = {x | BRx}
5	1, 2, 4	elab2 1901	1 |- (A e. [B]R <-> BRA)

Syntax hints:   <-> wb 146   e. wcel 958  Vcvv 1811   class class class wbr 2619  [cec 4259

This theorem is referenced by:  ecdmn0 4280  erthi 4281  erth 4282  erdisj 4286

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-ec 4263

Copyright terms: Public domain