Theorem elec 4263

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 2613	. 2 |- (x = A -> (BRx <-> BRA))
3		elec.2	. . 3 |- B e. V
4	3	dfec2 4248	. 2 |- [B]R = {x | BRx}
5	1, 2, 4	elab2 1892	1 |- (A e. [B]R <-> BRA)

Syntax hints:   <-> wb 146   e. wcel 955  Vcvv 1802   class class class wbr 2609  [cec 4243

This theorem is referenced by:  ecdmn0 4264  erthi 4265  erth 4266  erdisj 4270

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-ec 4247

Copyright terms: Public domain