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

Theorem alephf1ALT 7817
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 7779 . . 3  |-  aleph  Fn  On
2 alephon 7783 . . . . 5  |-  ( aleph `  x )  e.  On
32a1i 10 . . . 4  |-  ( x  e.  On  ->  ( aleph `  x )  e.  On )
43rgen 2684 . . 3  |-  A. x  e.  On  ( aleph `  x
)  e.  On
5 ffnfv 5765 . . 3  |-  ( aleph : On --> On  <->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x
)  e.  On ) )
61, 4, 5mpbir2an 886 . 2  |-  aleph : On --> On
7 alephsmo 7816 . 2  |-  Smo  aleph
8 smo11 6465 . 2  |-  ( (
aleph : On --> On  /\  Smo  aleph )  ->  aleph : On -1-1-> On )
96, 7, 8mp2an 653 1  |-  aleph : On -1-1-> On
Colors of variables: wff set class
Syntax hints:    e. wcel 1710   A.wral 2619   Oncon0 4471    Fn wfn 5329   -->wf 5330   -1-1->wf1 5331   ` cfv 5334   Smo wsmo 6446   alephcale 7656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-riota 6388  df-smo 6447  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-oi 7312  df-har 7359  df-card 7659  df-aleph 7660
  Copyright terms: Public domain W3C validator