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Theorem alephsmo 7988
Description: The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Assertion
Ref Expression
alephsmo  |-  Smo  aleph

Proof of Theorem alephsmo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3369 . 2  |-  On  C_  On
2 ordon 4766 . 2  |-  Ord  On
3 alephord2i 7963 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
43ralrimiv 2790 . . 3  |-  ( x  e.  On  ->  A. y  e.  x  ( aleph `  y )  e.  (
aleph `  x ) )
54rgen 2773 . 2  |-  A. x  e.  On  A. y  e.  x  ( aleph `  y
)  e.  ( aleph `  x )
6 alephfnon 7951 . . . 4  |-  aleph  Fn  On
7 alephsson 7986 . . . 4  |-  ran  aleph  C_  On
8 df-f 5461 . . . 4  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  ran  aleph  C_  On ) )
96, 7, 8mpbir2an 888 . . 3  |-  aleph : On --> On
10 issmo2 6614 . . 3  |-  ( aleph : On --> On  ->  (
( On  C_  On  /\ 
Ord  On  /\  A. x  e.  On  A. y  e.  x  ( aleph `  y
)  e.  ( aleph `  x ) )  ->  Smo  aleph ) )
119, 10ax-mp 5 . 2  |-  ( ( On  C_  On  /\  Ord  On 
/\  A. x  e.  On  A. y  e.  x  (
aleph `  y )  e.  ( aleph `  x )
)  ->  Smo  aleph )
121, 2, 5, 11mp3an 1280 1  |-  Smo  aleph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    e. wcel 1726   A.wral 2707    C_ wss 3322   Ord word 4583   Oncon0 4584   ran crn 4882    Fn wfn 5452   -->wf 5453   ` cfv 5457   Smo wsmo 6610   alephcale 7828
This theorem is referenced by:  alephf1ALT  7989  alephsing  8161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-smo 6611  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-oi 7482  df-har 7529  df-card 7831  df-aleph 7832
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