Theorem aleph0 4835

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 4789	. . 3 |- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
2	1	fveq1i 3710	. 2 |- (aleph` (/)) = (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/))
3		omex 4599	. . 3 |- om e. V
4	3	rdg0 3926	. 2 |- (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/)) = om
5	2, 4	eqtr 1487	1 |- (aleph` (/)) = om

Syntax hints:   = wceq 953  {crab 1640  (/)c0 2270  |^|cint 2523   class class class wbr 2609  {copab 2656  Oncon0 2938  omcom 3121  ` cfv 3172  reccrdg 3916   ~< csdm 4350  alephcale 4786

This theorem is referenced by:  alephon 4837  alephcard 4839  alephgeom 4854  cardaleph 4857  aleph1re 7494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-aleph 4789

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