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Theorem alephordi 8236
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 2499 . . 3  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 fveq2 5686 . . . 4  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32breq2d 4299 . . 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 2499 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 fveq2 5686 . . . 4  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
76breq2d 4299 . . 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 2499 . . 3  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 fveq2 5686 . . . 4  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
1110breq2d 4299 . . 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 2499 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 fveq2 5686 . . . 4  |-  ( x  =  B  ->  ( aleph `  x )  =  ( aleph `  B )
)
1514breq2d 4299 . . 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 3636 . . 3  |-  -.  A  e.  (/)
1817pm2.21i 131 . 2  |-  ( A  e.  (/)  ->  ( aleph `  A )  ~<  ( aleph `  (/) ) )
19 vex 2970 . . . . 5  |-  y  e. 
_V
2019elsuc2 4784 . . . 4  |-  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) )
21 alephordilem1 8235 . . . . . . . . 9  |-  ( y  e.  On  ->  ( aleph `  y )  ~< 
( aleph `  suc  y ) )
22 sdomtr 7441 . . . . . . . . 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 5686 . . . . . . . . 9  |-  ( A  =  y  ->  ( aleph `  A )  =  ( aleph `  y )
)
2827breq1d 4297 . . . . . . . 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 5696 . . . . . . 7  |-  ( aleph `  x )  e.  _V
3635a1i 11 . . . . . 6  |-  ( Lim  x  ->  ( aleph `  x )  e.  _V )
37 fveq2 5686 . . . . . . . 8  |-  ( w  =  A  ->  ( aleph `  w )  =  ( aleph `  A )
)
3837ssiun2s 4209 . . . . . . 7  |-  ( A  e.  x  ->  ( aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) )
39 vex 2970 . . . . . . . . 9  |-  x  e. 
_V
40 alephlim 8229 . . . . . . . . 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 3379 . . . . . . 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 7347 . . . . . 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 6455 . . . . . . . . . 10  |-  ( Lim  x  ->  ( A  e.  x  <->  suc  A  e.  x
) )
47 fveq2 5686 . . . . . . . . . . . . 13  |-  ( w  =  suc  A  -> 
( aleph `  w )  =  ( aleph `  suc  A ) )
4847ssiun2s 4209 . . . . . . . . . . . 12  |-  ( suc 
A  e.  x  -> 
( aleph `  suc  A ) 
C_  U_ w  e.  x  ( aleph `  w )
)
4941sseq2d 3379 . . . . . . . . . . . 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 7347 . . . . . . . . . . 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 7429 . . . . . . . 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 4777 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  Lim  x )  ->  x  e.  On )
5839, 57mpan 670 . . . . . . . . 9  |-  ( Lim  x  ->  x  e.  On )
59 onelon 4739 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A  e.  x )  ->  A  e.  On )
6058, 59sylan 471 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  A  e.  On )
61 ensym 7350 . . . . . . . . 9  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~~  ( aleph `  A )
)
62 alephordilem1 8235 . . . . . . . . 9  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
63 ensdomtr 7439 . . . . . . . . . 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 7324 . . . 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 6465 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 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    C_ wss 3323   (/)c0 3632   U_ciun 4166   class class class wbr 4287   Oncon0 4714   Lim wlim 4715   suc csuc 4716   ` cfv 5413    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301   alephcale 8098
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-om 6472  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-oi 7716  df-har 7765  df-card 8101  df-aleph 8102
This theorem is referenced by:  alephord  8237  alephval2  8728
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