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Theorem alephordi 7717
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 2357 . . 3  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 fveq2 5541 . . . 4  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32breq2d 4051 . . 3  |-  ( x  =  (/)  ->  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  (/) ) ) )
41, 3imbi12d 311 . 2  |-  ( x  =  (/)  ->  ( ( A  e.  x  -> 
( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  (/)  ->  ( aleph `  A
)  ~<  ( aleph `  (/) ) ) ) )
5 eleq2 2357 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 fveq2 5541 . . . 4  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
76breq2d 4051 . . 3  |-  ( x  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  y )
) )
85, 7imbi12d 311 . 2  |-  ( x  =  y  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) ) ) )
9 eleq2 2357 . . 3  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 fveq2 5541 . . . 4  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
1110breq2d 4051 . . 3  |-  ( x  =  suc  y  -> 
( ( aleph `  A
)  ~<  ( aleph `  x
)  <->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) )
129, 11imbi12d 311 . 2  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) )  <-> 
( A  e.  suc  y  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
13 eleq2 2357 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 fveq2 5541 . . . 4  |-  ( x  =  B  ->  ( aleph `  x )  =  ( aleph `  B )
)
1514breq2d 4051 . . 3  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
1613, 15imbi12d 311 . 2  |-  ( x  =  B  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B ) ) ) )
17 noel 3472 . . 3  |-  -.  A  e.  (/)
1817pm2.21i 123 . 2  |-  ( A  e.  (/)  ->  ( aleph `  A )  ~<  ( aleph `  (/) ) )
19 vex 2804 . . . . 5  |-  y  e. 
_V
2019elsuc2 4478 . . . 4  |-  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) )
21 alephordilem1 7716 . . . . . . . . 9  |-  ( y  e.  On  ->  ( aleph `  y )  ~< 
( aleph `  suc  y ) )
22 sdomtr 7015 . . . . . . . . 9  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  ( aleph `  y )  ~<  ( aleph `  suc  y ) )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) )
2321, 22sylan2 460 . . . . . . . 8  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  y  e.  On )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) )
2423expcom 424 . . . . . . 7  |-  ( y  e.  On  ->  (
( aleph `  A )  ~<  ( aleph `  y )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) )
2524imim2d 48 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
2625com23 72 . . . . 5  |-  ( y  e.  On  ->  ( A  e.  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
27 fveq2 5541 . . . . . . . . 9  |-  ( A  =  y  ->  ( aleph `  A )  =  ( aleph `  y )
)
2827breq1d 4049 . . . . . . . 8  |-  ( A  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  suc  y )  <-> 
( aleph `  y )  ~<  ( aleph `  suc  y ) ) )
2921, 28syl5ibr 212 . . . . . . 7  |-  ( A  =  y  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) )
3029a1d 22 . . . . . 6  |-  ( A  =  y  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  (
y  e.  On  ->  (
aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3130com3r 73 . . . . 5  |-  ( y  e.  On  ->  ( A  =  y  ->  ( ( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( aleph `  A )  ~< 
( aleph `  suc  y ) ) ) )
3226, 31jaod 369 . . . 4  |-  ( y  e.  On  ->  (
( A  e.  y  \/  A  =  y )  ->  ( ( A  e.  y  ->  (
aleph `  A )  ~< 
( aleph `  y )
)  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
3320, 32syl5bi 208 . . 3  |-  ( y  e.  On  ->  ( A  e.  suc  y  -> 
( ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
3433com23 72 . 2  |-  ( y  e.  On  ->  (
( A  e.  y  ->  ( aleph `  A
)  ~<  ( aleph `  y
) )  ->  ( A  e.  suc  y  -> 
( aleph `  A )  ~<  ( aleph `  suc  y ) ) ) )
35 fvex 5555 . . . . . 6  |-  ( aleph `  x )  e.  _V
36 fveq2 5541 . . . . . . . 8  |-  ( w  =  A  ->  ( aleph `  w )  =  ( aleph `  A )
)
3736ssiun2s 3962 . . . . . . 7  |-  ( A  e.  x  ->  ( aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) )
38 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
39 alephlim 7710 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ w  e.  x  ( aleph `  w )
)
4038, 39mpan 651 . . . . . . . 8  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ w  e.  x  ( aleph `  w ) )
4140sseq2d 3219 . . . . . . 7  |-  ( Lim  x  ->  ( ( aleph `  A )  C_  ( aleph `  x )  <->  (
aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) ) )
4237, 41syl5ibr 212 . . . . . 6  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  C_  ( aleph `  x ) ) )
43 ssdomg 6923 . . . . . 6  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  x )  -> 
( aleph `  A )  ~<_  ( aleph `  x )
) )
4435, 42, 43ee02 1367 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<_  ( aleph `  x ) ) )
45 limsuc 4656 . . . . . . . . . 10  |-  ( Lim  x  ->  ( A  e.  x  <->  suc  A  e.  x
) )
46 fveq2 5541 . . . . . . . . . . . . 13  |-  ( w  =  suc  A  -> 
( aleph `  w )  =  ( aleph `  suc  A ) )
4746ssiun2s 3962 . . . . . . . . . . . 12  |-  ( suc 
A  e.  x  -> 
( aleph `  suc  A ) 
C_  U_ w  e.  x  ( aleph `  w )
)
4840sseq2d 3219 . . . . . . . . . . . 12  |-  ( Lim  x  ->  ( ( aleph `  suc  A ) 
C_  ( aleph `  x
)  <->  ( aleph `  suc  A )  C_  U_ w  e.  x  ( aleph `  w
) ) )
4947, 48syl5ibr 212 . . . . . . . . . . 11  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A ) 
C_  ( aleph `  x
) ) )
50 ssdomg 6923 . . . . . . . . . . 11  |-  ( (
aleph `  x )  e. 
_V  ->  ( ( aleph ` 
suc  A )  C_  ( aleph `  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5135, 49, 50ee02 1367 . . . . . . . . . 10  |-  ( Lim  x  ->  ( suc  A  e.  x  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
) )
5245, 51sylbid 206 . . . . . . . . 9  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph ` 
suc  A )  ~<_  (
aleph `  x ) ) )
5352imp 418 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
)
54 domnsym 7003 . . . . . . . 8  |-  ( (
aleph `  suc  A )  ~<_  ( aleph `  x )  ->  -.  ( aleph `  x
)  ~<  ( aleph `  suc  A ) )
5553, 54syl 15 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
56 limelon 4471 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  Lim  x )  ->  x  e.  On )
5738, 56mpan 651 . . . . . . . . 9  |-  ( Lim  x  ->  x  e.  On )
58 onelon 4433 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A  e.  x )  ->  A  e.  On )
5957, 58sylan 457 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  A  e.  On )
60 ensym 6926 . . . . . . . . 9  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~~  ( aleph `  A )
)
61 alephordilem1 7716 . . . . . . . . 9  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
62 ensdomtr 7013 . . . . . . . . . 10  |-  ( ( ( aleph `  x )  ~~  ( aleph `  A )  /\  ( aleph `  A )  ~<  ( aleph `  suc  A ) )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
6362ex 423 . . . . . . . . 9  |-  ( (
aleph `  x )  ~~  ( aleph `  A )  ->  ( ( aleph `  A
)  ~<  ( aleph `  suc  A )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6460, 61, 63syl2im 34 . . . . . . . 8  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( A  e.  On  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6559, 64syl5com 26 . . . . . . 7  |-  ( ( Lim  x  /\  A  e.  x )  ->  (
( aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) ) )
6655, 65mtod 168 . . . . . 6  |-  ( ( Lim  x  /\  A  e.  x )  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
)
6766ex 423 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
) )
6844, 67jcad 519 . . . 4  |-  ( Lim  x  ->  ( A  e.  x  ->  ( (
aleph `  A )  ~<_  (
aleph `  x )  /\  -.  ( aleph `  A )  ~~  ( aleph `  x )
) ) )
69 brsdom 6900 . . . 4  |-  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  ( ( aleph `  A )  ~<_  ( aleph `  x )  /\  -.  ( aleph `  A
)  ~~  ( aleph `  x ) ) )
7068, 69syl6ibr 218 . . 3  |-  ( Lim  x  ->  ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) ) )
7170a1d 22 . 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 4666 1  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U_ciun 3921   class class class wbr 4039   Oncon0 4408   Lim wlim 4409   suc csuc 4410   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878   alephcale 7585
This theorem is referenced by:  alephord  7718  alephval2  8210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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