New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  intmin Unicode version

Theorem intmin 3946
 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.
StepHypRef Expression
1 vex 2862 . . . . 5
21elintrab 3938 . . . 4
3 ssid 3290 . . . . 5
4 sseq2 3293 . . . . . . 7
5 eleq2 2414 . . . . . . 7
64, 5imbi12d 311 . . . . . 6
76rspcv 2951 . . . . 5
83, 7mpii 39 . . . 4
92, 8syl5bi 208 . . 3
109ssrdv 3278 . 2
11 ssintub 3944 . . 3
1211a1i 10 . 2
1310, 12eqssd 3289 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1642   wcel 1710  wral 2614  crab 2618   wss 3257  cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by:  intmin2  3953
 Copyright terms: Public domain W3C validator