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Theorem alephsmo 7972
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 3359 . 2  |-  On  C_  On
2 ordon 4754 . 2  |-  Ord  On
3 alephord2i 7947 . . . 4  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( aleph `  y )  e.  ( aleph `  x )
) )
43ralrimiv 2780 . . 3  |-  ( x  e.  On  ->  A. y  e.  x  ( aleph `  y )  e.  (
aleph `  x ) )
54rgen 2763 . 2  |-  A. x  e.  On  A. y  e.  x  ( aleph `  y
)  e.  ( aleph `  x )
6 alephfnon 7935 . . . 4  |-  aleph  Fn  On
7 alephsson 7970 . . . 4  |-  ran  aleph  C_  On
8 df-f 5449 . . . 4  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  ran  aleph  C_  On ) )
96, 7, 8mpbir2an 887 . . 3  |-  aleph : On --> On
10 issmo2 6602 . . 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 2697    C_ wss 3312   Ord word 4572   Oncon0 4573   ran crn 4870    Fn wfn 5440   -->wf 5441   ` cfv 5445   Smo wsmo 6598   alephcale 7812
This theorem is referenced by:  alephf1ALT  7973  alephsing  8145
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-smo 6599  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816
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