Theorem aleph0 4863

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 4817	. . 3 |- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
2	1	fveq1i 3725	. 2 |- (aleph` (/)) = (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/))
3		omex 4627	. . 3 |- om e. V
4	3	rdg0 3941	. 2 |- (rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)` (/)) = om
5	2, 4	eqtr 1495	1 |- (aleph` (/)) = om

Syntax hints:   = wceq 956  {crab 1648  (/)c0 2280  |^|cint 2533   class class class wbr 2619  {copab 2666  Oncon0 2948  omcom 3131  ` cfv 3182  reccrdg 3931   ~< csdm 4366  alephcale 4814

This theorem is referenced by:  alephon 4865  alephcard 4867  alephgeom 4882  cardaleph 4885  aleph1re 7551

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-aleph 4817

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