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Theorem alephordi 8368
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 2455 . . 3  |-  ( x  =  (/)  ->  ( A  e.  x  <->  A  e.  (/) ) )
2 fveq2 5774 . . . 4  |-  ( x  =  (/)  ->  ( aleph `  x )  =  (
aleph `  (/) ) )
32breq2d 4379 . . 3  |-  ( x  =  (/)  ->  ( (
aleph `  A )  ~< 
( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  (/) ) ) )
41, 3imbi12d 318 . 2  |-  ( x  =  (/)  ->  ( ( A  e.  x  -> 
( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  (/)  ->  ( aleph `  A
)  ~<  ( aleph `  (/) ) ) ) )
5 eleq2 2455 . . 3  |-  ( x  =  y  ->  ( A  e.  x  <->  A  e.  y ) )
6 fveq2 5774 . . . 4  |-  ( x  =  y  ->  ( aleph `  x )  =  ( aleph `  y )
)
76breq2d 4379 . . 3  |-  ( x  =  y  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  y )
) )
85, 7imbi12d 318 . 2  |-  ( x  =  y  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  y  ->  ( aleph `  A )  ~<  ( aleph `  y ) ) ) )
9 eleq2 2455 . . 3  |-  ( x  =  suc  y  -> 
( A  e.  x  <->  A  e.  suc  y ) )
10 fveq2 5774 . . . 4  |-  ( x  =  suc  y  -> 
( aleph `  x )  =  ( aleph `  suc  y ) )
1110breq2d 4379 . . 3  |-  ( x  =  suc  y  -> 
( ( aleph `  A
)  ~<  ( aleph `  x
)  <->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) )
129, 11imbi12d 318 . 2  |-  ( x  =  suc  y  -> 
( ( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x ) )  <-> 
( A  e.  suc  y  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) ) ) )
13 eleq2 2455 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
14 fveq2 5774 . . . 4  |-  ( x  =  B  ->  ( aleph `  x )  =  ( aleph `  B )
)
1514breq2d 4379 . . 3  |-  ( x  =  B  ->  (
( aleph `  A )  ~<  ( aleph `  x )  <->  (
aleph `  A )  ~< 
( aleph `  B )
) )
1613, 15imbi12d 318 . 2  |-  ( x  =  B  ->  (
( A  e.  x  ->  ( aleph `  A )  ~<  ( aleph `  x )
)  <->  ( A  e.  B  ->  ( aleph `  A )  ~<  ( aleph `  B ) ) ) )
17 noel 3715 . . 3  |-  -.  A  e.  (/)
1817pm2.21i 131 . 2  |-  ( A  e.  (/)  ->  ( aleph `  A )  ~<  ( aleph `  (/) ) )
19 vex 3037 . . . . 5  |-  y  e. 
_V
2019elsuc2 4862 . . . 4  |-  ( A  e.  suc  y  <->  ( A  e.  y  \/  A  =  y ) )
21 alephordilem1 8367 . . . . . . . . 9  |-  ( y  e.  On  ->  ( aleph `  y )  ~< 
( aleph `  suc  y ) )
22 sdomtr 7574 . . . . . . . . 9  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  ( aleph `  y )  ~<  ( aleph `  suc  y ) )  ->  ( aleph `  A )  ~<  ( aleph `  suc  y ) )
2321, 22sylan2 472 . . . . . . . 8  |-  ( ( ( aleph `  A )  ~<  ( aleph `  y )  /\  y  e.  On )  ->  ( aleph `  A
)  ~<  ( aleph `  suc  y ) )
2423expcom 433 . . . . . . 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 5774 . . . . . . . . 9  |-  ( A  =  y  ->  ( aleph `  A )  =  ( aleph `  y )
)
2827breq1d 4377 . . . . . . . 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 378 . . . 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 5784 . . . . . . 7  |-  ( aleph `  x )  e.  _V
3635a1i 11 . . . . . 6  |-  ( Lim  x  ->  ( aleph `  x )  e.  _V )
37 fveq2 5774 . . . . . . . 8  |-  ( w  =  A  ->  ( aleph `  w )  =  ( aleph `  A )
)
3837ssiun2s 4287 . . . . . . 7  |-  ( A  e.  x  ->  ( aleph `  A )  C_  U_ w  e.  x  (
aleph `  w ) )
39 vex 3037 . . . . . . . . 9  |-  x  e. 
_V
40 alephlim 8361 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( aleph `  x )  = 
U_ w  e.  x  ( aleph `  w )
)
4139, 40mpan 668 . . . . . . . 8  |-  ( Lim  x  ->  ( aleph `  x )  =  U_ w  e.  x  ( aleph `  w ) )
4241sseq2d 3445 . . . . . . 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 7480 . . . . . 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 6583 . . . . . . . . . 10  |-  ( Lim  x  ->  ( A  e.  x  <->  suc  A  e.  x
) )
47 fveq2 5774 . . . . . . . . . . . . 13  |-  ( w  =  suc  A  -> 
( aleph `  w )  =  ( aleph `  suc  A ) )
4847ssiun2s 4287 . . . . . . . . . . . 12  |-  ( suc 
A  e.  x  -> 
( aleph `  suc  A ) 
C_  U_ w  e.  x  ( aleph `  w )
)
4941sseq2d 3445 . . . . . . . . . . . 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 7480 . . . . . . . . . . 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 427 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  ( aleph `  suc  A )  ~<_  ( aleph `  x )
)
55 domnsym 7562 . . . . . . . 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 4855 . . . . . . . . . 10  |-  ( ( x  e.  _V  /\  Lim  x )  ->  x  e.  On )
5839, 57mpan 668 . . . . . . . . 9  |-  ( Lim  x  ->  x  e.  On )
59 onelon 4817 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A  e.  x )  ->  A  e.  On )
6058, 59sylan 469 . . . . . . . 8  |-  ( ( Lim  x  /\  A  e.  x )  ->  A  e.  On )
61 ensym 7483 . . . . . . . . 9  |-  ( (
aleph `  A )  ~~  ( aleph `  x )  ->  ( aleph `  x )  ~~  ( aleph `  A )
)
62 alephordilem1 8367 . . . . . . . . 9  |-  ( A  e.  On  ->  ( aleph `  A )  ~< 
( aleph `  suc  A ) )
63 ensdomtr 7572 . . . . . . . . . 10  |-  ( ( ( aleph `  x )  ~~  ( aleph `  A )  /\  ( aleph `  A )  ~<  ( aleph `  suc  A ) )  ->  ( aleph `  x )  ~<  ( aleph `  suc  A ) )
6463ex 432 . . . . . . . . 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 432 . . . . 5  |-  ( Lim  x  ->  ( A  e.  x  ->  -.  ( aleph `  A )  ~~  ( aleph `  x )
) )
6945, 68jcad 531 . . . 4  |-  ( Lim  x  ->  ( A  e.  x  ->  ( (
aleph `  A )  ~<_  (
aleph `  x )  /\  -.  ( aleph `  A )  ~~  ( aleph `  x )
) ) )
70 brsdom 7457 . . . 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 6593 1  |-  ( B  e.  On  ->  ( A  e.  B  ->  (
aleph `  A )  ~< 
( aleph `  B )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    C_ wss 3389   (/)c0 3711   U_ciun 4243   class class class wbr 4367   Oncon0 4792   Lim wlim 4793   suc csuc 4794   ` cfv 5496    ~~ cen 7432    ~<_ cdom 7433    ~< csdm 7434   alephcale 8230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-oi 7850  df-har 7899  df-card 8233  df-aleph 8234
This theorem is referenced by:  alephord  8369  alephval2  8860
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