MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  meet0 Structured version   Unicode version

Theorem meet0 14566
Description: Lemma for odujoin 14571. (Contributed by Stefan O'Rear, 29-Jan-2015.)
Assertion
Ref Expression
meet0  |-  ( meet `  (/) )  =  (/)

Proof of Theorem meet0
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4341 . . 3  |-  (/)  e.  _V
2 base0 13508 . . . 4  |-  (/)  =  (
Base `  (/) )
3 eqid 2438 . . . 4  |-  ( glb `  (/) )  =  ( glb `  (/) )
4 eqid 2438 . . . 4  |-  ( meet `  (/) )  =  (
meet `  (/) )
52, 3, 4meetfval 14453 . . 3  |-  ( (/)  e.  _V  ->  ( meet `  (/) )  =  (
a  e.  (/) ,  b  e.  (/)  |->  ( ( glb `  (/) ) `  {
a ,  b } ) ) )
61, 5ax-mp 8 . 2  |-  ( meet `  (/) )  =  ( a  e.  (/) ,  b  e.  (/)  |->  ( ( glb `  (/) ) `  {
a ,  b } ) )
7 mpt20 6429 . 2  |-  ( a  e.  (/) ,  b  e.  (/)  |->  ( ( glb `  (/) ) `  {
a ,  b } ) )  =  (/)
86, 7eqtri 2458 1  |-  ( meet `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {cpr 3817   ` cfv 5456    e. cmpt2 6085   glbcglb 14402   meetcmee 14404
This theorem is referenced by:  odumeet  14569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-slot 13475  df-base 13476  df-meet 14436
  Copyright terms: Public domain W3C validator