Theorem intid 4710

Description: The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)

Hypothesis

Ref	Expression
intid.1

Assertion

Ref	Expression
intid

Distinct variable group:   ,

Proof of Theorem intid

Step	Hyp	Ref	Expression
1		snex 4693	. . 3
2		eleq2 2530	. . . 4
3		intid.1	. . . . 5
4	3	snid 4057	. . . 4
5	2, 4	intmin3 4315	. . 3
6	1, 5	ax-mp 5	. 2
7	3	elintab 4297	. . . 4
8		id 22	. . . 4
9	7, 8	mpgbir 1622	. . 3
10		snssi 4174	. . 3
11	9, 10	ax-mp 5	. 2
12	6, 11	eqssi 3519	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  {cab 2442  cvv 3109  C_wss 3475  {csn 4029  |^|cint 4286

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-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-int 4287