Theorem aleph0 3669

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 Kuratowski and Mostowski, Set Theory, 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."

Assertion

Ref	Expression
aleph0	⊢ (ℵ ‘∅) = ω

Proof of Theorem aleph0

Step	Hyp	Ref	Expression
1		df-aleph 3624	. . 3 ⊢ ℵ = rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω)
2	1	fveq1i 2833	. 2 ⊢ (ℵ ‘∅) = (rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω) ‘∅)
3		omex 3475	. . 3 ⊢ ω ∈ V
4	3	rdgzer 2979	. 2 ⊢ (rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω) ‘∅) = ω
5	2, 4	eqtr 1119	1 ⊢ (ℵ ‘∅) = ω

Syntax hints:   = wceq 1091  {crab 1204  ∅c0 1707  ∩cint 1965   class class class wbr 2054  {copab 2055  Oncon0 2199  ωcom 2372   ‘cfv 2422  reccrdg 2969   ≺ csdm 3273  ℵcale 3621

This theorem is referenced by:  alephon 3671  alephcard 3673  alephgeom 3687  cardaleph 3690

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-aleph 3624

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