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Theorem alephordi 8455
Description: Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephordi  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )

Proof of Theorem alephordi
Dummy variables  w  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2514 . . 3  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 fveq2 5853 . . . 4  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32breq2d 4446 . . 3  |-  ( x  =  (/)  ->  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  (/) ) ) )
41, 3imbi12d 320 . 2  |-  ( x  =  (/)  ->  ( ( A  e.  x  -> 
( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  (/)  ->  ( aleph `  A
)  ~<  ( aleph `  (/) ) ) ) )
5 eleq2 2514 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 fveq2 5853 . . . 4  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
76breq2d 4446 . . 3  |-  ( x  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  y )
) )
85, 7imbi12d 320 . 2  |-  ( x  =  y  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) ) ) )
9 eleq2 2514 . . 3  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 fveq2 5853 . . . 4  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
1110breq2d 4446 . . 3  |-  ( x  =  suc  y  -> 
( ( aleph `  A
)  ~<  ( aleph `  x
)  <->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) )
129, 11imbi12d 320 . 2  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) )  <-> 
( A  e.  suc  y  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
13 eleq2 2514 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 fveq2 5853 . . . 4  |-  ( x  =  B  ->  ( aleph `  x )  =  ( aleph `  B )
)
1514breq2d 4446 . . 3  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
1613, 15imbi12d 320 . 2  |-  ( x  =  B  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B ) ) ) )
17 noel 3772 . . 3  |-  -.  A  e.  (/)
1817pm2.21i 131 . 2  |-  ( A  e.  (/)  ->  ( aleph `  A )  ~<  ( aleph `  (/) ) )
19 vex 3096 . . . . 5  |-  y  e. 
_V
2019elsuc2 4935 . . . 4  |-  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) )
21 alephordilem1 8454 . . . . . . . . 9  |-  ( y  e.  On  ->  ( aleph `  y )  ~< 
( aleph `  suc  y ) )
22 sdomtr 7654 . . . . . . . . 9  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  ( aleph `  y )  ~<  ( aleph `  suc  y ) )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) )
2321, 22sylan2 474 . . . . . . . 8  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  y  e.  On )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) )
2423expcom 435 . . . . . . 7  |-  ( y  e.  On  ->  (
( aleph `  A )  ~<  ( aleph `  y )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) )
2524imim2d 52 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
2625com23 78 . . . . 5  |-  ( y  e.  On  ->  ( A  e.  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
27 fveq2 5853 . . . . . . . . 9  |-  ( A  =  y  ->  ( aleph `  A )  =  ( aleph `  y )
)
2827breq1d 4444 . . . . . . . 8  |-  ( A  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  suc  y )  <-> 
( aleph `  y )  ~<  ( aleph `  suc  y ) ) )
2921, 28syl5ibr 221 . . . . . . 7  |-  ( A  =  y  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) )
3029a1d 25 . . . . . 6  |-  ( A  =  y  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3130com3r 79 . . . . 5  |-  ( y  e.  On  ->  ( A  =  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3226, 31jaod 380 . . . 4  |-  ( y  e.  On  ->  (
( A  e.  y  \/  A  =  y )  ->  ( ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  y )
)  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
3320, 32syl5bi 217 . . 3  |-  ( y  e.  On  ->  ( A  e.  suc  y  -> 
( ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
3433com23 78 . 2  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  suc  y  -> 
( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
35 fvex 5863 . . . . . . 7  |-  ( aleph `  x )  e.  _V
3635a1i 11 . . . . . 6  |-  ( Lim  x  ->  ( aleph `  x )  e.  _V )
37 fveq2 5853 . . . . . . . 8  |-  ( w  =  A  ->  ( aleph `  w )  =  ( aleph `  A )
)
3837ssiun2s 4356 . . . . . . 7  |-  ( A  e.  x  ->  ( aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) )
39 vex 3096 . . . . . . . . 9  |-  x  e. 
_V
40 alephlim 8448 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ w  e.  x  ( aleph `  w )
)
4139, 40mpan 670 . . . . . . . 8  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ w  e.  x  ( aleph `  w ) )
4241sseq2d 3515 . . . . . . 7  |-  ( Lim  x  ->  ( ( aleph `  A )  C_  ( aleph `  x )  <->  (
aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) ) )
4338, 42syl5ibr 221 . . . . . 6  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  C_  ( aleph `  x ) ) )
44 ssdomg 7560 . . . . . 6  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  x )  -> 
( aleph `  A )  ~<_  ( aleph `  x )
) )
4536, 43, 44sylsyld 56 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<_  ( aleph `  x ) ) )
46 limsuc 6666 . . . . . . . . . 10  |-  ( Lim  x  ->  ( A  e.  x  <->  suc  A  e.  x
) )
47 fveq2 5853 . . . . . . . . . . . . 13  |-  ( w  =  suc  A  -> 
( aleph `  w )  =  ( aleph `  suc  A ) )
4847ssiun2s 4356 . . . . . . . . . . . 12  |-  ( suc 
A  e.  x  -> 
( aleph `  suc  A ) 
C_  U_ w  e.  x  ( aleph `  w )
)
4941sseq2d 3515 . . . . . . . . . . . 12  |-  ( Lim  x  ->  ( ( aleph `  suc  A ) 
C_  ( aleph `  x
)  <->  ( aleph `  suc  A )  C_  U_ w  e.  x  ( aleph `  w
) ) )
5048, 49syl5ibr 221 . . . . . . . . . . 11  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A ) 
C_  ( aleph `  x
) ) )
51 ssdomg 7560 . . . . . . . . . . 11  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph ` 
suc  A )  C_  ( aleph `  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5236, 50, 51sylsyld 56 . . . . . . . . . 10  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5346, 52sylbid 215 . . . . . . . . 9  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph ` 
suc  A )  ~<_  (
aleph `  x ) ) )
5453imp 429 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
)
55 domnsym 7642 . . . . . . . 8  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  x )  ->  -.  ( aleph `  x
)  ~<  ( aleph `  suc  A ) )
5654, 55syl 16 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
57 limelon 4928 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  Lim  x )  ->  x  e.  On )
5839, 57mpan 670 . . . . . . . . 9  |-  ( Lim  x  ->  x  e.  On )
59 onelon 4890 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A  e.  x )  ->  A  e.  On )
6058, 59sylan 471 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  A  e.  On )
61 ensym 7563 . . . . . . . . 9  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~~  ( aleph `  A )
)
62 alephordilem1 8454 . . . . . . . . 9  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
63 ensdomtr 7652 . . . . . . . . . 10  |-  ( ( ( aleph `  x )  ~~  ( aleph `  A )  /\  ( aleph `  A )  ~<  ( aleph `  suc  A ) )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
6463ex 434 . . . . . . . . 9  |-  ( (
aleph `  x )  ~~  ( aleph `  A )  ->  ( ( aleph `  A
)  ~<  ( aleph `  suc  A )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6561, 62, 64syl2im 38 . . . . . . . 8  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( A  e.  On  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6660, 65syl5com 30 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  (
( aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6756, 66mtod 177 . . . . . 6  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
)
6867ex 434 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
) )
6945, 68jcad 533 . . . 4  |-  ( Lim  x  ->  ( A  e.  x  ->  ( (
aleph `  A )  ~<_  (
aleph `  x )  /\  -.  ( aleph `  A )  ~~  ( aleph `  x )
) ) )
70 brsdom 7537 . . . 4  |-  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  ( ( aleph `  A )  ~<_  ( aleph `  x )  /\  -.  ( aleph `  A
)  ~~  ( aleph `  x ) ) )
7169, 70syl6ibr 227 . . 3  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) ) )
7271a1d 25 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  y )
)  ->  ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) ) ) )
734, 8, 12, 16, 18, 34, 72tfinds 6676 1  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093    C_ wss 3459   (/)c0 3768   U_ciun 4312   class class class wbr 4434   Oncon0 4865   Lim wlim 4866   suc csuc 4867   ` cfv 5575    ~~ cen 7512    ~<_ cdom 7513    ~< csdm 7514   alephcale 8317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-inf2 8058
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-se 4826  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-isom 5584  df-riota 6239  df-om 6683  df-recs 7041  df-rdg 7075  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-oi 7935  df-har 7984  df-card 8320  df-aleph 8321
This theorem is referenced by:  alephord  8456  alephval2  8947
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