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Theorem aleph0 7695
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 7575 . . 3  |-  aleph  =  rec (har ,  om )
21fveq1i 5528 . 2  |-  ( aleph `  (/) )  =  ( rec (har ,  om ) `  (/) )
3 omex 7346 . . 3  |-  om  e.  _V
43rdg0 6436 . 2  |-  ( rec (har ,  om ) `  (/) )  =  om
52, 4eqtri 2305 1  |-  ( aleph `  (/) )  =  om
Colors of variables: wff set class
Syntax hints:    = wceq 1625   (/)c0 3457   omcom 4658   ` cfv 5257   reccrdg 6424  harchar 7272   alephcale 7571
This theorem is referenced by:  alephon  7698  alephcard  7699  alephgeom  7711  cardaleph  7718  alephfplem1  7733  pwcfsdom  8207  alephom  8209  winalim2  8320  aleph1re  12525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-aleph 7575
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