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Theorem alephsmo 7745
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 3210 . 2  |-  On  C_  On
2 ordon 4590 . 2  |-  Ord  On
3 alephord2i 7720 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
43ralrimiv 2638 . . 3  |-  ( x  e.  On  ->  A. y  e.  x  ( aleph `  y )  e.  (
aleph `  x ) )
54rgen 2621 . 2  |-  A. x  e.  On  A. y  e.  x  ( aleph `  y
)  e.  ( aleph `  x )
6 alephfnon 7708 . . . 4  |-  aleph  Fn  On
7 alephsson 7743 . . . 4  |-  ran  aleph  C_  On
8 df-f 5275 . . . 4  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  ran  aleph  C_  On ) )
96, 7, 8mpbir2an 886 . . 3  |-  aleph : On --> On
10 issmo2 6382 . . 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 1277 1  |-  Smo  aleph
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1696   A.wral 2556    C_ wss 3165   Ord word 4407   Oncon0 4408   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271   Smo wsmo 6378   alephcale 7585
This theorem is referenced by:  alephf1ALT  7746  alephsing  7918
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-smo 6379  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-card 7588  df-aleph 7589
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