Theorem aleph0 10145

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 10021	. . 3 ⊢ ℵ = rec(har, ω)
2	1	fveq1i 6886	. 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅)
3		omex 9644	. . 3 ⊢ ω ∈ V
4	3	rdg0 8429	. 2 ⊢ (rec(har, ω)‘∅) = ω
5	2, 4	eqtri 2784	1 ⊢ (ℵ‘∅) = ω

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4279  ‘cfv 6538  ωcom 7877  reccrdg 8417  harchar 9550  ℵcale 10017

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-aleph 10021

This theorem is used by:  alephon  10148  alephcard  10149  alephgeom  10161  cardaleph  10168  alephfplem1  10183  pwcfsdom  10668  alephom  10670  winalim2  10781  aleph1re  16413  aleph1min  44557