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Theorem oneqmini 4938
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 4304 . . . . . 6  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
2 ssel 3493 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( A  e.  B  ->  A  e.  On ) )
3 ssel 3493 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( x  e.  B  ->  x  e.  On ) )
42, 3anim12d 563 . . . . . . . . . . 11  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  e.  On  /\  x  e.  On ) ) )
5 ontri1 4921 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  x  <->  -.  x  e.  A ) )
64, 5syl6 33 . . . . . . . . . 10  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  C_  x  <->  -.  x  e.  A ) ) )
76expdimp 437 . . . . . . . . 9  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
x  e.  B  -> 
( A  C_  x  <->  -.  x  e.  A ) ) )
87pm5.74d 247 . . . . . . . 8  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  B  ->  -.  x  e.  A
) ) )
9 con2b 334 . . . . . . . 8  |-  ( ( x  e.  B  ->  -.  x  e.  A
)  <->  ( x  e.  A  ->  -.  x  e.  B ) )
108, 9syl6bb 261 . . . . . . 7  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  A  ->  -.  x  e.  B
) ) )
1110ralbidv2 2892 . . . . . 6  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  B  A  C_  x  <->  A. x  e.  A  -.  x  e.  B ) )
121, 11syl5bb 257 . . . . 5  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A  C_  |^| B  <->  A. x  e.  A  -.  x  e.  B ) )
1312biimprd 223 . . . 4  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  A  -.  x  e.  B  ->  A  C_  |^| B ) )
1413expimpd 603 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  C_  |^| B ) )
15 intss1 4303 . . . . 5  |-  ( A  e.  B  ->  |^| B  C_  A )
1615a1i 11 . . . 4  |-  ( B 
C_  On  ->  ( A  e.  B  ->  |^| B  C_  A ) )
1716adantrd 468 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  |^| B  C_  A
) )
1814, 17jcad 533 . 2  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  ( A  C_  |^| B  /\  |^| B  C_  A
) ) )
19 eqss 3514 . 2  |-  ( A  =  |^| B  <->  ( A  C_ 
|^| B  /\  |^| B  C_  A ) )
2018, 19syl6ibr 227 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 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   |^|cint 4288   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  oneqmin  6639  alephval3  8508  cfsuc  8654  alephval2  8964
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