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| 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 9980 | . . 3 ⊢ ℵ = rec(har, ω) | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ (ℵ‘∅) = (rec(har, ω)‘∅) |
| 3 | omex 9683 | . . 3 ⊢ ω ∈ V | |
| 4 | 3 | rdg0 8461 | . 2 ⊢ (rec(har, ω)‘∅) = ω |
| 5 | 2, 4 | eqtri 2765 | 1 ⊢ (ℵ‘∅) = ω |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∅c0 4333 ‘cfv 6561 ωcom 7887 reccrdg 8449 harchar 9596 ℵcale 9976 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-aleph 9980 |
| This theorem is referenced by: alephon 10109 alephcard 10110 alephgeom 10122 cardaleph 10129 alephfplem1 10144 pwcfsdom 10623 alephom 10625 winalim2 10736 aleph1re 16281 aleph1min 43570 |
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