Theorem aleph0 4846

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 4800	. . 3 ⊢ ℵ = rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω)
2	1	fveq1i 3720	. 2 ⊢ (ℵ ‘∅) = (rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω) ‘∅)
3		omex 4610	. . 3 ⊢ ω ∈ V
4	3	rdg0 3936	. 2 ⊢ (rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω) ‘∅) = ω
5	2, 4	eqtr 1493	1 ⊢ (ℵ ‘∅) = ω

Syntax hints:   = wceq 955  {crab 1646  ∅c0 2277  ∩cint 2529   class class class wbr 2615  {copab 2662  Oncon0 2944  ωcom 3127   ‘cfv 3178  reccrdg 3926   ≺ csdm 4359  ℵcale 4797

This theorem is referenced by:  alephon 4848  alephcard 4850  alephgeom 4865  cardaleph 4868  aleph1re 7511

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-rdg 3927  df-aleph 4800

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