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Mirrors > Home > MPE Home > Th. List > alephsuc2 | Structured version Visualization version GIF version |
Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 9487 function by transfinite recursion, starting from Ο. Using this theorem we could define the aleph function with {π§ β On β£ π§ βΌ π₯} in place of β© {π§ β On β£ π₯ βΊ π§} in df-aleph 9883. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
alephsuc2 | β’ (π΄ β On β (β΅βsuc π΄) = {π₯ β On β£ π₯ βΌ (β΅βπ΄)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephon 10012 | . . . . . . 7 β’ (β΅βsuc π΄) β On | |
2 | 1 | oneli 6436 | . . . . . 6 β’ (π¦ β (β΅βsuc π΄) β π¦ β On) |
3 | alephcard 10013 | . . . . . . . . 9 β’ (cardβ(β΅βsuc π΄)) = (β΅βsuc π΄) | |
4 | iscard 9918 | . . . . . . . . 9 β’ ((cardβ(β΅βsuc π΄)) = (β΅βsuc π΄) β ((β΅βsuc π΄) β On β§ βπ¦ β (β΅βsuc π΄)π¦ βΊ (β΅βsuc π΄))) | |
5 | 3, 4 | mpbi 229 | . . . . . . . 8 β’ ((β΅βsuc π΄) β On β§ βπ¦ β (β΅βsuc π΄)π¦ βΊ (β΅βsuc π΄)) |
6 | 5 | simpri 487 | . . . . . . 7 β’ βπ¦ β (β΅βsuc π΄)π¦ βΊ (β΅βsuc π΄) |
7 | 6 | rspec 3236 | . . . . . 6 β’ (π¦ β (β΅βsuc π΄) β π¦ βΊ (β΅βsuc π΄)) |
8 | 2, 7 | jca 513 | . . . . 5 β’ (π¦ β (β΅βsuc π΄) β (π¦ β On β§ π¦ βΊ (β΅βsuc π΄))) |
9 | sdomel 9075 | . . . . . . 7 β’ ((π¦ β On β§ (β΅βsuc π΄) β On) β (π¦ βΊ (β΅βsuc π΄) β π¦ β (β΅βsuc π΄))) | |
10 | 1, 9 | mpan2 690 | . . . . . 6 β’ (π¦ β On β (π¦ βΊ (β΅βsuc π΄) β π¦ β (β΅βsuc π΄))) |
11 | 10 | imp 408 | . . . . 5 β’ ((π¦ β On β§ π¦ βΊ (β΅βsuc π΄)) β π¦ β (β΅βsuc π΄)) |
12 | 8, 11 | impbii 208 | . . . 4 β’ (π¦ β (β΅βsuc π΄) β (π¦ β On β§ π¦ βΊ (β΅βsuc π΄))) |
13 | breq1 5113 | . . . . 5 β’ (π₯ = π¦ β (π₯ βΊ (β΅βsuc π΄) β π¦ βΊ (β΅βsuc π΄))) | |
14 | 13 | elrab 3650 | . . . 4 β’ (π¦ β {π₯ β On β£ π₯ βΊ (β΅βsuc π΄)} β (π¦ β On β§ π¦ βΊ (β΅βsuc π΄))) |
15 | 12, 14 | bitr4i 278 | . . 3 β’ (π¦ β (β΅βsuc π΄) β π¦ β {π₯ β On β£ π₯ βΊ (β΅βsuc π΄)}) |
16 | 15 | eqriv 2734 | . 2 β’ (β΅βsuc π΄) = {π₯ β On β£ π₯ βΊ (β΅βsuc π΄)} |
17 | alephsucdom 10022 | . . 3 β’ (π΄ β On β (π₯ βΌ (β΅βπ΄) β π₯ βΊ (β΅βsuc π΄))) | |
18 | 17 | rabbidv 3418 | . 2 β’ (π΄ β On β {π₯ β On β£ π₯ βΌ (β΅βπ΄)} = {π₯ β On β£ π₯ βΊ (β΅βsuc π΄)}) |
19 | 16, 18 | eqtr4id 2796 | 1 β’ (π΄ β On β (β΅βsuc π΄) = {π₯ β On β£ π₯ βΌ (β΅βπ΄)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 {crab 3410 class class class wbr 5110 Oncon0 6322 suc csuc 6324 βcfv 6501 βΌ cdom 8888 βΊ csdm 8889 cardccrd 9878 β΅cale 9879 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-oi 9453 df-har 9500 df-card 9882 df-aleph 9883 |
This theorem is referenced by: alephsuc3 10523 |
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