Theorem intirr 5390

Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
intirr

Distinct variable group:   ,

Proof of Theorem intirr

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		incom 3690	. . . 4
2	1	eqeq1i 2464	. . 3
3		disj2 3874	. . 3
4		reli 5135	. . . 4
5		ssrel 5096	. . . 4
6	4, 5	ax-mp 5	. . 3
7	2, 3, 6	3bitri 271	. 2
8		equcom 1794	. . . . 5
9		vex 3112	. . . . . 6
10	9	ideq 5160	. . . . 5
11		df-br 4453	. . . . 5
12	8, 10, 11	3bitr2i 273	. . . 4
13		opex 4716	. . . . . . 7
14	13	biantrur 506	. . . . . 6
15		eldif 3485	. . . . . 6
16	14, 15	bitr4i 252	. . . . 5
17		df-br 4453	. . . . 5
18	16, 17	xchnxbir 309	. . . 4
19	12, 18	imbi12i 326	. . 3
20	19	2albii 1641	. 2
21		nfv 1707	. . . 4
22		breq2 4456	. . . . 5
23	22	notbid 294	. . . 4
24	21, 23	equsal 2036	. . 3
25	24	albii 1640	. 2
26	7, 20, 25	3bitr2i 273	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  cvv 3109  \cdif 3472  i^icin 3474  C_wss 3475  c0 3784  <.cop 4035   class class class wbr 4452  cid 4795  Relwrel 5009

This theorem is referenced by:  hartogslem1  7988  hausdiag  20146

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-id 4800  df-xp 5010  df-rel 5011