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Theorem alephf1ALT 7976
Description: The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
alephf1ALT  |-  aleph : On -1-1-> On

Proof of Theorem alephf1ALT
StepHypRef Expression
1 alephfnon 7938 . . 3  |-  aleph  Fn  On
2 alephon 7942 . . . . 5  |-  ( aleph `  x )  e.  On
32a1i 11 . . . 4  |-  ( x  e.  On  ->  ( aleph `  x )  e.  On )
43rgen 2763 . . 3  |-  A. x  e.  On  ( aleph `  x
)  e.  On
5 ffnfv 5886 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x
)  e.  On ) )
61, 4, 5mpbir2an 887 . 2  |-  aleph : On --> On
7 alephsmo 7975 . 2  |-  Smo  aleph
8 smo11 6618 . 2  |-  ( (
aleph : On --> On  /\  Smo  aleph )  ->  aleph : On -1-1-> On )
96, 7, 8mp2an 654 1  |-  aleph : On -1-1-> On
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   A.wral 2697   Oncon0 4573    Fn wfn 5441   -->wf 5442   -1-1->wf1 5443   ` cfv 5446   Smo wsmo 6599   alephcale 7815
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 4693  ax-inf2 7588
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-smo 6600  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-card 7818  df-aleph 7819
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