Theorem intmin 4306

Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
intmin

Distinct variable groups:   ,   ,

Proof of Theorem intmin

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . . . 5
2	1	elintrab 4298	. . . 4
3		ssid 3522	. . . . 5
4		sseq2 3525	. . . . . . 7
5		eleq2 2530	. . . . . . 7
6	4, 5	imbi12d 320	. . . . . 6
7	6	rspcv 3206	. . . . 5
8	3, 7	mpii 43	. . . 4
9	2, 8	syl5bi 217	. . 3
10	9	ssrdv 3509	. 2
11		ssintub 4304	. . 3
12	11	a1i 11	. 2
13	10, 12	eqssd 3520	1

Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818  A.wral 2807  {crab 2811  C_wss 3475  |^|cint 4286

This theorem is referenced by:  intmin2  4314  ordintdif  4932  bm2.5ii  6641  onsucmin  6656  rankonidlem  8267  rankval4  8306  mrcid  15010  lspid  17628  aspid  17979  cldcls  19543  spanid  26265  chsupid  26330  igenidl2  30462  pclidN  35620  diaocN  36852

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

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-ral 2812  df-rab 2816  df-v 3111  df-in 3482  df-ss 3489  df-int 4287