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Theorem onint 4523
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
StepHypRef Expression
1 ordon 4511 . . . 4  |-  Ord  On
2 tz7.5 4350 . . . 4  |-  ( ( Ord  On  /\  A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
31, 2mp3an1 1269 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
4 ssel 3116 . . . . . . . . . . . . . . . 16  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
54imdistani 674 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A  C_  On  /\  x  e.  On ) )
6 ssel 3116 . . . . . . . . . . . . . . . . . . . 20  |-  ( A 
C_  On  ->  ( z  e.  A  ->  z  e.  On ) )
7 ontri1 4363 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( x  C_  z  <->  -.  z  e.  x ) )
8 ssel 3116 . . . . . . . . . . . . . . . . . . . . . 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 641 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  C_  On  /\  x  e.  On )  ->  (
z  e.  A  -> 
( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
1211com4r 82 . . . . . . . . . . . . . . . . . 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 2592 . . . . . . . . . . . . . . . 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 3437 . . . . . . . . . . . . . . . 16  |-  ( ( A  i^i  x )  =  (/)  <->  A. z  e.  A  -.  z  e.  x
)
16 vex 2743 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
1716elint2 3810 . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . 13  |-  ( y  e.  x  ->  ( A  C_  On  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A
) ) ) )
2120com4l 80 . . . . . . . . . . . 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 3127 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  C_  |^| A
)
24 intss1 3818 . . . . . . . . . . 11  |-  ( x  e.  A  ->  |^| A  C_  x )
2524ad2antrl 711 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  |^| A  C_  x
)
2623, 25eqssd 3138 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  =  |^| A )
2726eleq1d 2322 . . . . . . . 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 591 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
x  e.  A  ->  |^| A  e.  A ) ) ) )
3029com34 79 . . . . 5  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  |^| A  e.  A
) ) ) )
3130pm2.43d 46 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  |^| A  e.  A ) ) )
3231rexlimdv 2637 . . 3  |-  ( A 
C_  On  ->  ( E. x  e.  A  ( A  i^i  x )  =  (/)  ->  |^| A  e.  A ) )
333, 32syl5 30 . 2  |-  ( A 
C_  On  ->  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A ) )
3433anabsi5 793 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    i^i cin 3093    C_ wss 3094   (/)c0 3397   |^|cint 3803   Ord word 4328   Oncon0 4329
This theorem is referenced by:  onint0  4524  onssmin  4525  onminesb  4526  onminsb  4527  oninton  4528  oneqmin  4533  oeeulem  6532  nnawordex  6568  unblem1  7042  unblem2  7043  tz9.12lem3  7394  scott0  7489  cardid2  7519  ackbij1lem18  7796  cardcf  7811  cff1  7817  cflim2  7822  cfss  7824  cofsmo  7828  fin23lem26  7884  pwfseqlem3  8215  gruina  8373  2ndcdisj  17109  sltval2  23643  nocvxmin  23679  axfelem5  23684  rankeq1o  24141  dnnumch3  26476
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-tr 4054  df-eprel 4242  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333
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