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

Theorem oneqmini 5474
Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
Assertion
Ref Expression
oneqmini  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem oneqmini
StepHypRef Expression
1 ssint 4250 . . . . . 6  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
2 ssel 3426 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( A  e.  B  ->  A  e.  On ) )
3 ssel 3426 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( x  e.  B  ->  x  e.  On ) )
42, 3anim12d 566 . . . . . . . . . . 11  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  e.  On  /\  x  e.  On ) ) )
5 ontri1 5457 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  x  <->  -.  x  e.  A ) )
64, 5syl6 34 . . . . . . . . . 10  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  C_  x  <->  -.  x  e.  A ) ) )
76expdimp 439 . . . . . . . . 9  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
x  e.  B  -> 
( A  C_  x  <->  -.  x  e.  A ) ) )
87pm5.74d 251 . . . . . . . 8  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  B  ->  -.  x  e.  A
) ) )
9 con2b 336 . . . . . . . 8  |-  ( ( x  e.  B  ->  -.  x  e.  A
)  <->  ( x  e.  A  ->  -.  x  e.  B ) )
108, 9syl6bb 265 . . . . . . 7  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  A  ->  -.  x  e.  B
) ) )
1110ralbidv2 2823 . . . . . 6  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  B  A  C_  x  <->  A. x  e.  A  -.  x  e.  B ) )
121, 11syl5bb 261 . . . . 5  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A  C_  |^| B  <->  A. x  e.  A  -.  x  e.  B ) )
1312biimprd 227 . . . 4  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  A  -.  x  e.  B  ->  A  C_  |^| B ) )
1413expimpd 608 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  C_  |^| B ) )
15 intss1 4249 . . . . 5  |-  ( A  e.  B  ->  |^| B  C_  A )
1615a1i 11 . . . 4  |-  ( B 
C_  On  ->  ( A  e.  B  ->  |^| B  C_  A ) )
1716adantrd 470 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  |^| B  C_  A
) )
1814, 17jcad 536 . 2  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  ( A  C_  |^| B  /\  |^| B  C_  A
) ) )
19 eqss 3447 . 2  |-  ( A  =  |^| B  <->  ( A  C_ 
|^| B  /\  |^| B  C_  A ) )
2018, 19syl6ibr 231 1  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737    C_ wss 3404   |^|cint 4234   Oncon0 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426  df-on 5427
This theorem is referenced by:  oneqmin  6632  alephval3  8541  cfsuc  8687  alephval2  8997
  Copyright terms: Public domain W3C validator