MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onint Unicode version

Theorem onint 4586
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 4574 . . . 4  |-  Ord  On
2 tz7.5 4413 . . . 4  |-  ( ( Ord  On  /\  A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
31, 2mp3an1 1264 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  E. x  e.  A  ( A  i^i  x )  =  (/) )
4 ssel 3174 . . . . . . . . . . . . . . . 16  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
54imdistani 671 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  On  /\  x  e.  A )  ->  ( A  C_  On  /\  x  e.  On ) )
6 ssel 3174 . . . . . . . . . . . . . . . . . . . 20  |-  ( A 
C_  On  ->  ( z  e.  A  ->  z  e.  On ) )
7 ontri1 4426 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( x  C_  z  <->  -.  z  e.  x ) )
8 ssel 3174 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x 
C_  z  ->  (
y  e.  x  -> 
y  e.  z ) )
97, 8syl6bir 220 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  On  /\  z  e.  On )  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) )
109ex 423 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  On  ->  (
z  e.  On  ->  ( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
116, 10sylan9 638 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  C_  On  /\  x  e.  On )  ->  (
z  e.  A  -> 
( -.  z  e.  x  ->  ( y  e.  x  ->  y  e.  z ) ) ) )
1211com4r 80 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  x  ->  (
( A  C_  On  /\  x  e.  On )  ->  ( z  e.  A  ->  ( -.  z  e.  x  ->  y  e.  z ) ) ) )
1312imp31 421 . . . . . . . . . . . . . . . . 17  |-  ( ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On ) )  /\  z  e.  A )  ->  ( -.  z  e.  x  ->  y  e.  z ) )
1413ralimdva 2621 . . . . . . . . . . . . . . . 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 3495 . . . . . . . . . . . . . . . 16  |-  ( ( A  i^i  x )  =  (/)  <->  A. z  e.  A  -.  z  e.  x
)
16 vex 2791 . . . . . . . . . . . . . . . . 17  |-  y  e. 
_V
1716elint2 3869 . . . . . . . . . . . . . . . 16  |-  ( y  e.  |^| A  <->  A. z  e.  A  y  e.  z )
1814, 15, 173imtr4g 261 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  On )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
195, 18sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  ( A  C_  On  /\  x  e.  A )
)  ->  ( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A ) )
2019exp32 588 . . . . . . . . . . . . 13  |-  ( y  e.  x  ->  ( A  C_  On  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  y  e.  |^| A
) ) ) )
2120com4l 78 . . . . . . . . . . . 12  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
y  e.  x  -> 
y  e.  |^| A
) ) ) )
2221imp32 422 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( y  e.  x  ->  y  e.  |^| A ) )
2322ssrdv 3185 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  C_  |^| A
)
24 intss1 3877 . . . . . . . . . . 11  |-  ( x  e.  A  ->  |^| A  C_  x )
2524ad2antrl 708 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  |^| A  C_  x
)
2623, 25eqssd 3196 . . . . . . . . 9  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  x  =  |^| A )
2726eleq1d 2349 . . . . . . . 8  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  <->  |^| A  e.  A
) )
2827biimpd 198 . . . . . . 7  |-  ( ( A  C_  On  /\  (
x  e.  A  /\  ( A  i^i  x
)  =  (/) ) )  ->  ( x  e.  A  ->  |^| A  e.  A ) )
2928exp32 588 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  (
x  e.  A  ->  |^| A  e.  A ) ) ) )
3029com34 77 . . . . 5  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
x  e.  A  -> 
( ( A  i^i  x )  =  (/)  ->  |^| A  e.  A
) ) ) )
3130pm2.43d 44 . . . 4  |-  ( A 
C_  On  ->  ( x  e.  A  ->  (
( A  i^i  x
)  =  (/)  ->  |^| A  e.  A ) ) )
3231rexlimdv 2666 . . 3  |-  ( A 
C_  On  ->  ( E. x  e.  A  ( A  i^i  x )  =  (/)  ->  |^| A  e.  A ) )
333, 32syl5 28 . 2  |-  ( A 
C_  On  ->  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A ) )
3433anabsi5 790 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   |^|cint 3862   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onint0  4587  onssmin  4588  onminesb  4589  onminsb  4590  oninton  4591  oneqmin  4596  oeeulem  6599  nnawordex  6635  unblem1  7109  unblem2  7110  tz9.12lem3  7461  scott0  7556  cardid2  7586  ackbij1lem18  7863  cardcf  7878  cff1  7884  cflim2  7889  cfss  7891  cofsmo  7895  fin23lem26  7951  pwfseqlem3  8282  gruina  8440  2ndcdisj  17182  sltval2  24310  nocvxmin  24345  nobndlem5  24350  rankeq1o  24801  dnnumch3  27144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
  Copyright terms: Public domain W3C validator