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Theorem alephf1ALT 7944
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 7906 . . 3  |-  aleph  Fn  On
2 alephon 7910 . . . . 5  |-  ( aleph `  x )  e.  On
32a1i 11 . . . 4  |-  ( x  e.  On  ->  ( aleph `  x )  e.  On )
43rgen 2735 . . 3  |-  A. x  e.  On  ( aleph `  x
)  e.  On
5 ffnfv 5857 . . 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 7943 . 2  |-  Smo  aleph
8 smo11 6589 . 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 1721   A.wral 2670   Oncon0 4545    Fn wfn 5412   -->wf 5413   -1-1->wf1 5414   ` cfv 5417   Smo wsmo 6570   alephcale 7783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-smo 6571  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-har 7486  df-card 7786  df-aleph 7787
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