Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqintg Unicode version

Theorem eqintg 25064
 Description: To prove that a set is the finest one that has the property prove that is a part of all sets that has this property and that has also that property. (Contributed by FL, 15-Oct-2012.)
Hypotheses
Ref Expression
eqintg.1
eqintg.2
eqintg.3
Assertion
Ref Expression
eqintg
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem eqintg
StepHypRef Expression
1 eqintg.3 . . . . . 6
21ex 423 . . . . 5
32alrimiv 1621 . . . 4
5 ssintab 3895 . . 3
64, 5sylibr 203 . 2
7 eqintg.2 . . . . 5
87adantr 451 . . . 4
9 simpr 447 . . . . 5
10 eqintg.1 . . . . . . 7
1110bicomd 192 . . . . . 6
1211ax-gen 1536 . . . . 5
13 elabgt 2924 . . . . 5
149, 12, 13sylancl 643 . . . 4
158, 14mpbird 223 . . 3
16 intss1 3893 . . 3
1715, 16syl 15 . 2
186, 17eqssd 3209 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358  wal 1530   wceq 1632   wcel 1696  cab 2282   wss 3165  cint 3878 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-ss 3179  df-int 3879
 Copyright terms: Public domain W3C validator