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Theorem onint 4777
Description: The intersection (infimum) of a non-empty 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 4765 . . . 4  |-  Ord  On
2 tz7.5 4604 . . . 4  |-  ( ( Ord  On  /\  A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
31, 2mp3an1 1267 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
4 ssel 3344 . . . . . . . . . . . . . . . 16  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
54imdistani 673 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A  C_  On  /\  x  e.  On ) )
6 ssel 3344 . . . . . . . . . . . . . . . . . . . 20  |-  ( A 
C_  On  ->  ( z  e.  A  ->  z  e.  On ) )
7 ontri1 4617 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( x  C_  z  <->  -.  z  e.  x ) )
8 ssel 3344 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x 
C_  z  ->  (
y  e.  x  -> 
y  e.  z ) )
97, 8syl6bir 222 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) )
109ex 425 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  (
z  e.  On  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
116, 10sylan9 640 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  C_  On  /\  x  e.  On )  ->  (
z  e.  A  -> 
( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
1211com4r 83 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  x  ->  (
( A  C_  On  /\  x  e.  On )  ->  ( z  e.  A  ->  ( -.  z  e.  x  ->  y  e.  z ) ) ) )
1312imp31 423 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On ) )  /\  z  e.  A )  ->  ( -.  z  e.  x  ->  y  e.  z ) )
1413ralimdva 2786 . . . . . . . . . . . . . . . 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 3670 . . . . . . . . . . . . . . . 16  |-  ( ( A  i^i  x )  =  (/)  <->  A. z  e.  A  -.  z  e.  x
)
16 vex 2961 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
1716elint2 4059 . . . . . . . . . . . . . . . 16  |-  ( y  e.  |^| A  <->  A. z  e.  A  y  e.  z )
1814, 15, 173imtr4g 263 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
195, 18sylan2 462 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  A )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
2019exp32 590 . . . . . . . . . . . . 13  |-  ( y  e.  x  ->  ( A  C_  On  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A
) ) ) )
2120com4l 81 . . . . . . . . . . . 12  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
y  e.  x  -> 
y  e.  |^| A
) ) ) )
2221imp32 424 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( y  e.  x  ->  y  e.  |^| A ) )
2322ssrdv 3356 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  C_  |^| A
)
24 intss1 4067 . . . . . . . . . . 11  |-  ( x  e.  A  ->  |^| A  C_  x )
2524ad2antrl 710 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  |^| A  C_  x
)
2623, 25eqssd 3367 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  =  |^| A )
2726eleq1d 2504 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  <->  |^| A  e.  A
) )
2827biimpd 200 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  ->  |^| A  e.  A ) )
2928exp32 590 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
x  e.  A  ->  |^| A  e.  A ) ) ) )
3029com34 80 . . . . 5  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  |^| A  e.  A
) ) ) )
3130pm2.43d 47 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  |^| A  e.  A ) ) )
3231rexlimdv 2831 . . 3  |-  ( A 
C_  On  ->  ( E. x  e.  A  ( A  i^i  x )  =  (/)  ->  |^| A  e.  A ) )
333, 32syl5 31 . 2  |-  ( A 
C_  On  ->  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A ) )
3433anabsi5 792 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   (/)c0 3630   |^|cint 4052   Ord word 4582   Oncon0 4583
This theorem is referenced by:  onint0  4778  onssmin  4779  onminesb  4780  onminsb  4781  oninton  4782  oneqmin  4787  oeeulem  6846  nnawordex  6882  unblem1  7361  unblem2  7362  tz9.12lem3  7717  scott0  7812  cardid2  7842  ackbij1lem18  8119  cardcf  8134  cff1  8140  cflim2  8145  cfss  8147  cofsmo  8151  fin23lem26  8207  pwfseqlem3  8537  gruina  8695  2ndcdisj  17521  sltval2  25613  nocvxmin  25648  nobndlem5  25653  rankeq1o  26114  dnnumch3  27124
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-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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