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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 9498, 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 8371 | . 2 β’ (π΄ β On β (rec(har, Ο)βsuc π΄) = (harβ(rec(har, Ο)βπ΄))) | |
2 | df-aleph 9881 | . . 3 β’ β΅ = rec(har, Ο) | |
3 | 2 | fveq1i 6844 | . 2 β’ (β΅βsuc π΄) = (rec(har, Ο)βsuc π΄) |
4 | 2 | fveq1i 6844 | . . 3 β’ (β΅βπ΄) = (rec(har, Ο)βπ΄) |
5 | 4 | fveq2i 6846 | . 2 β’ (harβ(β΅βπ΄)) = (harβ(rec(har, Ο)βπ΄)) |
6 | 1, 3, 5 | 3eqtr4g 2798 | 1 β’ (π΄ β On β (β΅βsuc π΄) = (harβ(β΅βπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Oncon0 6318 suc csuc 6320 βcfv 6497 Οcom 7803 reccrdg 8356 harchar 9497 β΅cale 9877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-aleph 9881 |
This theorem is referenced by: alephon 10010 alephcard 10011 alephnbtwn 10012 alephordilem1 10014 cardaleph 10030 gchaleph2 10613 aleph1min 41917 |
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