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Theorem oneqmini 4871
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 4245 . . . . . 6  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
2 ssel 3451 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( A  e.  B  ->  A  e.  On ) )
3 ssel 3451 . . . . . . . . . . . 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 4854 . . . . . . . . . . 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 2830 . . . . . 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 4244 . . . . 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 3472 . 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 1370    e. wcel 1758   A.wral 2795    C_ wss 3429   |^|cint 4229   Oncon0 4820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824
This theorem is referenced by:  oneqmin  6519  alephval3  8384  cfsuc  8530  alephval2  8840
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