Theorem aleph0 4881

Description: The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers om (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 aleph₀. 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	|- (aleph` (/)) = om

Proof of Theorem aleph0

Step	Hyp	Ref	Expression
1		df-aleph 4834	. . 3 |- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
2	1	fveq1i 3732	. 2 |- (aleph` (/)) = (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/))
3		omex 4643	. . 3 |- om e. V
4	3	rdg0 3948	. 2 |- (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/)) = om
5	2, 4	eqtr 1498	1 |- (aleph` (/)) = om

Syntax hints:   = wceq 958  {crab 1651  (/)c0 2284  |^|cint 2538   class class class wbr 2625  {copab 2672  Oncon0 2955  omcom 3138  ` cfv 3189  reccrdg 3938   ~< csdm 4373  alephcale 4831

This theorem is referenced by:  alephon 4883  alephcard 4885  alephgeom 4900  cardaleph 4903  aleph1re 7559

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-inf2 4641

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-if 2367  df-pw 2407  df-sn 2417  df-pr 2418  df-tp 2420  df-op 2421  df-uni 2509  df-iun 2573  df-br 2626  df-opab 2673  df-tr 2687  df-eprel 2839  df-id 2842  df-po 2847  df-so 2857  df-fr 2924  df-we 2941  df-ord 2958  df-on 2959  df-lim 2960  df-suc 2961  df-om 3139  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-fv 3205  df-rdg 3939  df-aleph 4834

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