MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aleph0 Structured version   Unicode version

Theorem aleph0 8399
Description: The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers  om (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written 
aleph_0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Moshé Machover, Set Theory, Logic, and Their Limitations, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
aleph0  |-  ( aleph `  (/) )  =  om

Proof of Theorem aleph0
StepHypRef Expression
1 df-aleph 8273 . . 3  |-  aleph  =  rec (har ,  om )
21fveq1i 5806 . 2  |-  ( aleph `  (/) )  =  ( rec (har ,  om ) `  (/) )
3 omex 8013 . . 3  |-  om  e.  _V
43rdg0 7044 . 2  |-  ( rec (har ,  om ) `  (/) )  =  om
52, 4eqtri 2431 1  |-  ( aleph `  (/) )  =  om
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   (/)c0 3737   ` cfv 5525   omcom 6638   reccrdg 7032  harchar 7936   alephcale 8269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-om 6639  df-recs 6999  df-rdg 7033  df-aleph 8273
This theorem is referenced by:  alephon  8402  alephcard  8403  alephgeom  8415  cardaleph  8422  alephfplem1  8437  pwcfsdom  8910  alephom  8912  winalim2  9024  aleph1re  14079
  Copyright terms: Public domain W3C validator