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Theorem alephordi 7889
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 2449 . . 3  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 fveq2 5669 . . . 4  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32breq2d 4166 . . 3  |-  ( x  =  (/)  ->  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  (/) ) ) )
41, 3imbi12d 312 . 2  |-  ( x  =  (/)  ->  ( ( A  e.  x  -> 
( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  (/)  ->  ( aleph `  A
)  ~<  ( aleph `  (/) ) ) ) )
5 eleq2 2449 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 fveq2 5669 . . . 4  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
76breq2d 4166 . . 3  |-  ( x  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  y )
) )
85, 7imbi12d 312 . 2  |-  ( x  =  y  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) ) ) )
9 eleq2 2449 . . 3  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 fveq2 5669 . . . 4  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
1110breq2d 4166 . . 3  |-  ( x  =  suc  y  -> 
( ( aleph `  A
)  ~<  ( aleph `  x
)  <->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) )
129, 11imbi12d 312 . 2  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) )  <-> 
( A  e.  suc  y  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
13 eleq2 2449 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 fveq2 5669 . . . 4  |-  ( x  =  B  ->  ( aleph `  x )  =  ( aleph `  B )
)
1514breq2d 4166 . . 3  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
1613, 15imbi12d 312 . 2  |-  ( x  =  B  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B ) ) ) )
17 noel 3576 . . 3  |-  -.  A  e.  (/)
1817pm2.21i 125 . 2  |-  ( A  e.  (/)  ->  ( aleph `  A )  ~<  ( aleph `  (/) ) )
19 vex 2903 . . . . 5  |-  y  e. 
_V
2019elsuc2 4593 . . . 4  |-  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) )
21 alephordilem1 7888 . . . . . . . . 9  |-  ( y  e.  On  ->  ( aleph `  y )  ~< 
( aleph `  suc  y ) )
22 sdomtr 7182 . . . . . . . . 9  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  ( aleph `  y )  ~<  ( aleph `  suc  y ) )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) )
2321, 22sylan2 461 . . . . . . . 8  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  y  e.  On )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) )
2423expcom 425 . . . . . . 7  |-  ( y  e.  On  ->  (
( aleph `  A )  ~<  ( aleph `  y )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) )
2524imim2d 50 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
2625com23 74 . . . . 5  |-  ( y  e.  On  ->  ( A  e.  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
27 fveq2 5669 . . . . . . . . 9  |-  ( A  =  y  ->  ( aleph `  A )  =  ( aleph `  y )
)
2827breq1d 4164 . . . . . . . 8  |-  ( A  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  suc  y )  <-> 
( aleph `  y )  ~<  ( aleph `  suc  y ) ) )
2921, 28syl5ibr 213 . . . . . . 7  |-  ( A  =  y  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) )
3029a1d 23 . . . . . 6  |-  ( A  =  y  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3130com3r 75 . . . . 5  |-  ( y  e.  On  ->  ( A  =  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3226, 31jaod 370 . . . 4  |-  ( y  e.  On  ->  (
( A  e.  y  \/  A  =  y )  ->  ( ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  y )
)  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
3320, 32syl5bi 209 . . 3  |-  ( y  e.  On  ->  ( A  e.  suc  y  -> 
( ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
3433com23 74 . 2  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  suc  y  -> 
( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
35 fvex 5683 . . . . . 6  |-  ( aleph `  x )  e.  _V
36 fveq2 5669 . . . . . . . 8  |-  ( w  =  A  ->  ( aleph `  w )  =  ( aleph `  A )
)
3736ssiun2s 4077 . . . . . . 7  |-  ( A  e.  x  ->  ( aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) )
38 vex 2903 . . . . . . . . 9  |-  x  e. 
_V
39 alephlim 7882 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ w  e.  x  ( aleph `  w )
)
4038, 39mpan 652 . . . . . . . 8  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ w  e.  x  ( aleph `  w ) )
4140sseq2d 3320 . . . . . . 7  |-  ( Lim  x  ->  ( ( aleph `  A )  C_  ( aleph `  x )  <->  (
aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) ) )
4237, 41syl5ibr 213 . . . . . 6  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  C_  ( aleph `  x ) ) )
43 ssdomg 7090 . . . . . 6  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  x )  -> 
( aleph `  A )  ~<_  ( aleph `  x )
) )
4435, 42, 43ee02 1383 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<_  ( aleph `  x ) ) )
45 limsuc 4770 . . . . . . . . . 10  |-  ( Lim  x  ->  ( A  e.  x  <->  suc  A  e.  x
) )
46 fveq2 5669 . . . . . . . . . . . . 13  |-  ( w  =  suc  A  -> 
( aleph `  w )  =  ( aleph `  suc  A ) )
4746ssiun2s 4077 . . . . . . . . . . . 12  |-  ( suc 
A  e.  x  -> 
( aleph `  suc  A ) 
C_  U_ w  e.  x  ( aleph `  w )
)
4840sseq2d 3320 . . . . . . . . . . . 12  |-  ( Lim  x  ->  ( ( aleph `  suc  A ) 
C_  ( aleph `  x
)  <->  ( aleph `  suc  A )  C_  U_ w  e.  x  ( aleph `  w
) ) )
4947, 48syl5ibr 213 . . . . . . . . . . 11  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A ) 
C_  ( aleph `  x
) ) )
50 ssdomg 7090 . . . . . . . . . . 11  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph ` 
suc  A )  C_  ( aleph `  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5135, 49, 50ee02 1383 . . . . . . . . . 10  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5245, 51sylbid 207 . . . . . . . . 9  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph ` 
suc  A )  ~<_  (
aleph `  x ) ) )
5352imp 419 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
)
54 domnsym 7170 . . . . . . . 8  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  x )  ->  -.  ( aleph `  x
)  ~<  ( aleph `  suc  A ) )
5553, 54syl 16 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
56 limelon 4586 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  Lim  x )  ->  x  e.  On )
5738, 56mpan 652 . . . . . . . . 9  |-  ( Lim  x  ->  x  e.  On )
58 onelon 4548 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A  e.  x )  ->  A  e.  On )
5957, 58sylan 458 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  A  e.  On )
60 ensym 7093 . . . . . . . . 9  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~~  ( aleph `  A )
)
61 alephordilem1 7888 . . . . . . . . 9  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
62 ensdomtr 7180 . . . . . . . . . 10  |-  ( ( ( aleph `  x )  ~~  ( aleph `  A )  /\  ( aleph `  A )  ~<  ( aleph `  suc  A ) )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
6362ex 424 . . . . . . . . 9  |-  ( (
aleph `  x )  ~~  ( aleph `  A )  ->  ( ( aleph `  A
)  ~<  ( aleph `  suc  A )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6460, 61, 63syl2im 36 . . . . . . . 8  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( A  e.  On  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6559, 64syl5com 28 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  (
( aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6655, 65mtod 170 . . . . . 6  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
)
6766ex 424 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
) )
6844, 67jcad 520 . . . 4  |-  ( Lim  x  ->  ( A  e.  x  ->  ( (
aleph `  A )  ~<_  (
aleph `  x )  /\  -.  ( aleph `  A )  ~~  ( aleph `  x )
) ) )
69 brsdom 7067 . . . 4  |-  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  ( ( aleph `  A )  ~<_  ( aleph `  x )  /\  -.  ( aleph `  A
)  ~~  ( aleph `  x ) ) )
7068, 69syl6ibr 219 . . 3  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) ) )
7170a1d 23 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  y )
)  ->  ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) ) ) )
724, 8, 12, 16, 18, 34, 71tfinds 4780 1  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   _Vcvv 2900    C_ wss 3264   (/)c0 3572   U_ciun 4036   class class class wbr 4154   Oncon0 4523   Lim wlim 4524   suc csuc 4525   ` cfv 5395    ~~ cen 7043    ~<_ cdom 7044    ~< csdm 7045   alephcale 7757
This theorem is referenced by:  alephord  7890  alephval2  8381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-oi 7413  df-har 7460  df-card 7760  df-aleph 7761
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