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Theorem alephsmo 7983
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 3367 . 2  |-  On  C_  On
2 ordon 4763 . 2  |-  Ord  On
3 alephord2i 7958 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
43ralrimiv 2788 . . 3  |-  ( x  e.  On  ->  A. y  e.  x  ( aleph `  y )  e.  (
aleph `  x ) )
54rgen 2771 . 2  |-  A. x  e.  On  A. y  e.  x  ( aleph `  y
)  e.  ( aleph `  x )
6 alephfnon 7946 . . . 4  |-  aleph  Fn  On
7 alephsson 7981 . . . 4  |-  ran  aleph  C_  On
8 df-f 5458 . . . 4  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  ran  aleph  C_  On ) )
96, 7, 8mpbir2an 887 . . 3  |-  aleph : On --> On
10 issmo2 6611 . . 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 8 . 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 1279 1  |-  Smo  aleph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1725   A.wral 2705    C_ wss 3320   Ord word 4580   Oncon0 4581   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454   Smo wsmo 6607   alephcale 7823
This theorem is referenced by:  alephf1ALT  7984  alephsing  8156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-smo 6608  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-card 7826  df-aleph 7827
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