Theorem iserd 7356

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
iserd.1
iserd.2
iserd.3
iserd.4

Assertion

Ref	Expression
iserd

Distinct variable groups:   ,,,   ,   ,,,

Proof of Theorem iserd

Step	Hyp	Ref	Expression
1		iserd.1	. . 3
2		eqidd 2458	. . 3
3		iserd.2	. . . . . . . 8
4	3	ex 434	. . . . . . 7
5		iserd.3	. . . . . . . 8
6	5	ex 434	. . . . . . 7
7	4, 6	jca 532	. . . . . 6
8	7	alrimiv 1719	. . . . 5
9	8	alrimiv 1719	. . . 4
10	9	alrimiv 1719	. . 3
11		dfer2 7331	. . 3
12	1, 2, 10, 11	syl3anbrc 1180	. 2
13	12	adantr 465	. . . . . . . 8
14		simpr 461	. . . . . . . 8
15	13, 14	erref 7350	. . . . . . 7
16	15	ex 434	. . . . . 6
17		vex 3112	. . . . . . 7
18	17, 17	breldm 5212	. . . . . 6
19	16, 18	impbid1 203	. . . . 5
20		iserd.4	. . . . 5
21	19, 20	bitr4d 256	. . . 4
22	21	eqrdv 2454	. . 3
23		ereq2 7338	. . 3
24	22, 23	syl 16	. 2
25	12, 24	mpbid 210	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818   class class class wbr 4452  domcdm 5004  Relwrel 5009  Erwer 7327

This theorem is referenced by:  swoer  7358  eqer  7363  0er  7365  iiner  7402  erinxp  7404  ecopover  7434  ener  7582  eqger  16251  gicer  16324  gaorber  16346  efgrelexlemb  16768  efgcpbllemb  16773  hmpher  20285  xmeter  20936  phtpcer  21495  vitalilem1  22017  ercgrg  23908  erclwwlk  24816  erclwwlkn  24828  metider  27873  cicer  32590

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  df-co 5013  df-dm 5014  df-er 7330