Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > aleph0 | Structured version Visualization version GIF version |
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 Moshé Machover, Set Theory, Logic, and Their Limitations, 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." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
aleph0 | ⊢ (ℵ‘∅) = ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-aleph 9443 | . . 3 ⊢ ℵ = rec(har, ω) | |
2 | 1 | fveq1i 6676 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
3 | omex 9180 | . . 3 ⊢ ω ∈ V | |
4 | 3 | rdg0 8087 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
5 | 2, 4 | eqtri 2761 | 1 ⊢ (ℵ‘∅) = ω |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∅c0 4212 ‘cfv 6340 ωcom 7600 reccrdg 8075 harchar 9094 ℵcale 9439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7480 ax-inf2 9178 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-om 7601 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-aleph 9443 |
This theorem is referenced by: alephon 9570 alephcard 9571 alephgeom 9583 cardaleph 9590 alephfplem1 9605 pwcfsdom 10084 alephom 10086 winalim2 10197 aleph1re 15691 aleph1min 40701 |
Copyright terms: Public domain | W3C validator |