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Mirrors > Home > MPE Home > Th. List > alephsuc | Structured version Visualization version GIF version |
Description: Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 9451, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
alephsuc | ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsuc 8362 | . 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴))) | |
2 | df-aleph 9834 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fveq1i 6840 | . 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴) |
4 | 2 | fveq1i 6840 | . . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴) |
5 | 4 | fveq2i 6842 | . 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴)) |
6 | 1, 3, 5 | 3eqtr4g 2801 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Oncon0 6315 suc csuc 6317 ‘cfv 6493 ωcom 7794 reccrdg 8347 harchar 9450 ℵcale 9830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-aleph 9834 |
This theorem is referenced by: alephon 9963 alephcard 9964 alephnbtwn 9965 alephordilem1 9967 cardaleph 9983 gchaleph2 10566 aleph1min 41734 |
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