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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 9588, 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 | β’ (π΄ β On β (β΅βsuc π΄) = (harβ(β΅βπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsuc 8451 | . 2 β’ (π΄ β On β (rec(har, Ο)βsuc π΄) = (harβ(rec(har, Ο)βπ΄))) | |
2 | df-aleph 9971 | . . 3 β’ β΅ = rec(har, Ο) | |
3 | 2 | fveq1i 6903 | . 2 β’ (β΅βsuc π΄) = (rec(har, Ο)βsuc π΄) |
4 | 2 | fveq1i 6903 | . . 3 β’ (β΅βπ΄) = (rec(har, Ο)βπ΄) |
5 | 4 | fveq2i 6905 | . 2 β’ (harβ(β΅βπ΄)) = (harβ(rec(har, Ο)βπ΄)) |
6 | 1, 3, 5 | 3eqtr4g 2793 | 1 β’ (π΄ β On β (β΅βsuc π΄) = (harβ(β΅βπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Oncon0 6374 suc csuc 6376 βcfv 6553 Οcom 7876 reccrdg 8436 harchar 9587 β΅cale 9967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-aleph 9971 |
This theorem is referenced by: alephon 10100 alephcard 10101 alephnbtwn 10102 alephordilem1 10104 cardaleph 10120 gchaleph2 10703 aleph1min 43018 |
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