MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephf1ALT Structured version   Unicode version

Theorem alephf1ALT 8015
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 7977 . . 3  |-  aleph  Fn  On
2 alephon 7981 . . . . 5  |-  ( aleph `  x )  e.  On
32a1i 11 . . . 4  |-  ( x  e.  On  ->  ( aleph `  x )  e.  On )
43rgen 2777 . . 3  |-  A. x  e.  On  ( aleph `  x
)  e.  On
5 ffnfv 5923 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x
)  e.  On ) )
61, 4, 5mpbir2an 888 . 2  |-  aleph : On --> On
7 alephsmo 8014 . 2  |-  Smo  aleph
8 smo11 6655 . 2  |-  ( (
aleph : On --> On  /\  Smo  aleph )  ->  aleph : On -1-1-> On )
96, 7, 8mp2an 655 1  |-  aleph : On -1-1-> On
Colors of variables: wff set class
Syntax hints:    e. wcel 1727   A.wral 2711   Oncon0 4610    Fn wfn 5478   -->wf 5479   -1-1->wf1 5480   ` cfv 5483   Smo wsmo 6636   alephcale 7854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-riota 6578  df-smo 6637  df-recs 6662  df-rdg 6697  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-har 7555  df-card 7857  df-aleph 7858
  Copyright terms: Public domain W3C validator