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Theorem aleph0 8351
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 8225 . . 3  |-  aleph  =  rec (har ,  om )
21fveq1i 5803 . 2  |-  ( aleph `  (/) )  =  ( rec (har ,  om ) `  (/) )
3 omex 7964 . . 3  |-  om  e.  _V
43rdg0 6990 . 2  |-  ( rec (har ,  om ) `  (/) )  =  om
52, 4eqtri 2483 1  |-  ( aleph `  (/) )  =  om
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   (/)c0 3748   ` cfv 5529   omcom 6589   reccrdg 6978  harchar 7886   alephcale 8221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-aleph 8225
This theorem is referenced by:  alephon  8354  alephcard  8355  alephgeom  8367  cardaleph  8374  alephfplem1  8389  pwcfsdom  8862  alephom  8864  winalim2  8978  aleph1re  13649
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