Theorem elirrv 8044

Description: The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 8050 and efrirr 4865, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)

Assertion

Ref	Expression
elirrv

Proof of Theorem elirrv

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2529	. . . 4
2		ssnid 4058	. . . 4
3	1, 2	spei 2012	. . 3
4		snex 4693	. . . 4
5	4	zfregcl 8041	. . 3
6	3, 5	ax-mp 5	. 2
7		elsn 4043	. . . . . . 7
8		ax-9 1822	. . . . . . . . 9
9	8	equcoms 1795	. . . . . . . 8
10	9	com12 31	. . . . . . 7
11	7, 10	syl5bi 217	. . . . . 6
12		eleq1 2529	. . . . . . . . 9
13	12	notbid 294	. . . . . . . 8
14	13	rspccv 3207	. . . . . . 7
15	2, 14	mt2i 118	. . . . . 6
16	11, 15	nsyli 141	. . . . 5
17	16	con2d 115	. . . 4
18	17	ralrimiv 2869	. . 3
19		ralnex 2903	. . 3
20	18, 19	sylib 196	. 2
21	6, 20	mt2 179	1

Syntax hints:  -.wn 3  ->wi 4  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  {csn 4029

This theorem is referenced by:  elirr  8045  ruv  8048  dfac2  8532  nd1  8983  nd2  8984  nd3  8985  axunnd  8992  axregndlem1  9000  axregndlem2  9001  axregnd  9002  axregndOLD  9003  elpotr  29213  exnel  29235  distel  29236

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  ax-reg 8039

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032