Theorem aleph0 10057

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 9933	. . 3 ⊢ ℵ = rec(har, ω)
2	1	fveq1i 6882	. 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅)
3		omex 9610	. . 3 ⊢ ω ∈ V
4	3	rdg0 8406	. 2 ⊢ (rec(har, ω)‘∅) = ω
5	2, 4	eqtri 2785	1 ⊢ (ℵ‘∅) = ω

This proof depends on syntax axioms:   = wceq 1569  ∅c0 4285  ‘cfv 6536  ωcom 7860  reccrdg 8394  harchar 9516  ℵcale 9929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-aleph 9933

This theorem is used by:  alephon  10060  alephcard  10061  alephgeom  10073  cardaleph  10080  alephfplem1  10095  pwcfsdom  10574  alephom  10576  winalim2  10687  aleph1re  16307  aleph1min  44311