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Theorem aleph0 7661
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 7541 . . 3  |-  aleph  =  rec (har ,  om )
21fveq1i 5459 . 2  |-  ( aleph `  (/) )  =  ( rec (har ,  om ) `  (/) )
3 omex 7312 . . 3  |-  om  e.  _V
43rdg0 6402 . 2  |-  ( rec (har ,  om ) `  (/) )  =  om
52, 4eqtri 2278 1  |-  ( aleph `  (/) )  =  om
Colors of variables: wff set class
Syntax hints:    = wceq 1619   (/)c0 3430   omcom 4628   ` cfv 4673   reccrdg 6390  harchar 7238   alephcale 7537
This theorem is referenced by:  alephon  7664  alephcard  7665  alephgeom  7677  cardaleph  7684  alephfplem1  7699  pwcfsdom  8173  alephom  8175  winalim2  8286  aleph1re  12486
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484  ax-inf2 7310
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-aleph 7541
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