Theorem aleph0 10109

Description: The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers ω (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 ℵ₀. 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	⊢ (ℵ‘∅) = ω

Proof of Theorem aleph0

Step	Hyp	Ref	Expression
1		df-aleph 9983	. . 3 ⊢ ℵ = rec(har, ω)
2	1	fveq1i 6902	. 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅)
3		omex 9686	. . 3 ⊢ ω ∈ V
4	3	rdg0 8451	. 2 ⊢ (rec(har, ω)‘∅) = ω
5	2, 4	eqtri 2754	1 ⊢ (ℵ‘∅) = ω

Syntax hints:   = wceq 1534  ∅c0 4325  ‘cfv 6554  ωcom 7876  reccrdg 8439  harchar 9599  ℵcale 9979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746  ax-inf2 9684

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-aleph 9983

This theorem is referenced by:  alephon  10112  alephcard  10113  alephgeom  10125  cardaleph  10132  alephfplem1  10147  pwcfsdom  10626  alephom  10628  winalim2  10739  aleph1re  16247  aleph1min  43224

Otomatik - 173.255.232.114 CloudFlare DNS Türk Telekom DNS Google DNS Open DNS

OSZAR »