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Theorem onint 6427
Description: The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)
Assertion
Ref Expression
onint  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )

Proof of Theorem onint
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordon 6415 . . . 4  |-  Ord  On
2 tz7.5 4761 . . . 4  |-  ( ( Ord  On  /\  A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
31, 2mp3an1 1301 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
4 ssel 3371 . . . . . . . . . . . . . . . 16  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
54imdistani 690 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A  C_  On  /\  x  e.  On ) )
6 ssel 3371 . . . . . . . . . . . . . . . . . . . 20  |-  ( A 
C_  On  ->  ( z  e.  A  ->  z  e.  On ) )
7 ontri1 4774 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( x  C_  z  <->  -.  z  e.  x ) )
8 ssel 3371 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x 
C_  z  ->  (
y  e.  x  -> 
y  e.  z ) )
97, 8syl6bir 229 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) )
109ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  (
z  e.  On  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
116, 10sylan9 657 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  C_  On  /\  x  e.  On )  ->  (
z  e.  A  -> 
( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
1211com4r 86 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  x  ->  (
( A  C_  On  /\  x  e.  On )  ->  ( z  e.  A  ->  ( -.  z  e.  x  ->  y  e.  z ) ) ) )
1312imp31 432 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On ) )  /\  z  e.  A )  ->  ( -.  z  e.  x  ->  y  e.  z ) )
1413ralimdva 2815 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On )
)  ->  ( A. z  e.  A  -.  z  e.  x  ->  A. z  e.  A  y  e.  z ) )
15 disj 3740 . . . . . . . . . . . . . . . 16  |-  ( ( A  i^i  x )  =  (/)  <->  A. z  e.  A  -.  z  e.  x
)
16 vex 2996 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
1716elint2 4156 . . . . . . . . . . . . . . . 16  |-  ( y  e.  |^| A  <->  A. z  e.  A  y  e.  z )
1814, 15, 173imtr4g 270 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
195, 18sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  A )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
2019exp32 605 . . . . . . . . . . . . 13  |-  ( y  e.  x  ->  ( A  C_  On  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A
) ) ) )
2120com4l 84 . . . . . . . . . . . 12  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
y  e.  x  -> 
y  e.  |^| A
) ) ) )
2221imp32 433 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( y  e.  x  ->  y  e.  |^| A ) )
2322ssrdv 3383 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  C_  |^| A
)
24 intss1 4164 . . . . . . . . . . 11  |-  ( x  e.  A  ->  |^| A  C_  x )
2524ad2antrl 727 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  |^| A  C_  x
)
2623, 25eqssd 3394 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  =  |^| A )
2726eleq1d 2509 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  <->  |^| A  e.  A
) )
2827biimpd 207 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  ->  |^| A  e.  A ) )
2928exp32 605 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
x  e.  A  ->  |^| A  e.  A ) ) ) )
3029com34 83 . . . . 5  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  |^| A  e.  A
) ) ) )
3130pm2.43d 48 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  |^| A  e.  A ) ) )
3231rexlimdv 2861 . . 3  |-  ( A 
C_  On  ->  ( E. x  e.  A  ( A  i^i  x )  =  (/)  ->  |^| A  e.  A ) )
333, 32syl5 32 . 2  |-  ( A 
C_  On  ->  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A ) )
3433anabsi5 813 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737    i^i cin 3348    C_ wss 3349   (/)c0 3658   |^|cint 4149   Ord word 4739   Oncon0 4740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744
This theorem is referenced by:  onint0  6428  onssmin  6429  onminesb  6430  onminsb  6431  oninton  6432  oneqmin  6437  oeeulem  7061  nnawordex  7097  unblem1  7585  unblem2  7586  tz9.12lem3  8017  scott0  8114  cardid2  8144  ackbij1lem18  8427  cardcf  8442  cff1  8448  cflim2  8453  cfss  8455  cofsmo  8459  fin23lem26  8515  pwfseqlem3  8848  gruina  9006  2ndcdisj  19082  sltval2  27819  nocvxmin  27854  nobndlem5  27859  rankeq1o  28231  dnnumch3  29426
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