Theorem cnvsym 5386

Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvsym

Distinct variable group:   ,,

Proof of Theorem cnvsym

Step	Hyp	Ref	Expression
1		alcom 1845	. 2
2		relcnv 5379	. . 3
3		ssrel 5096	. . 3
4	2, 3	ax-mp 5	. 2
5		vex 3112	. . . . . 6
6		vex 3112	. . . . . 6
7	5, 6	brcnv 5190	. . . . 5
8		df-br 4453	. . . . 5
9	7, 8	bitr3i 251	. . . 4
10		df-br 4453	. . . 4
11	9, 10	imbi12i 326	. . 3
12	11	2albii 1641	. 2
13	1, 4, 12	3bitr4i 277	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  e.wcel 1818  C_wss 3475  <.cop 4035   class class class wbr 4452  `'ccnv 5003  Relwrel 5009

This theorem is referenced by:  dfer2  7331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012