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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 9173, 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 8160 | . 2 ⊢ (𝐴 ∈ On → (rec(har, ω)‘suc 𝐴) = (har‘(rec(har, ω)‘𝐴))) | |
2 | df-aleph 9556 | . . 3 ⊢ ℵ = rec(har, ω) | |
3 | 2 | fveq1i 6718 | . 2 ⊢ (ℵ‘suc 𝐴) = (rec(har, ω)‘suc 𝐴) |
4 | 2 | fveq1i 6718 | . . 3 ⊢ (ℵ‘𝐴) = (rec(har, ω)‘𝐴) |
5 | 4 | fveq2i 6720 | . 2 ⊢ (har‘(ℵ‘𝐴)) = (har‘(rec(har, ω)‘𝐴)) |
6 | 1, 3, 5 | 3eqtr4g 2803 | 1 ⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 Oncon0 6213 suc csuc 6215 ‘cfv 6380 ωcom 7644 reccrdg 8145 harchar 9172 ℵcale 9552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-aleph 9556 |
This theorem is referenced by: alephon 9683 alephcard 9684 alephnbtwn 9685 alephordilem1 9687 cardaleph 9703 gchaleph2 10286 aleph1min 40840 |
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