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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 9541 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 9937. (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 10066 | . . . . . . 7 β’ (β΅βsuc π΄) β On | |
2 | 1 | oneli 6472 | . . . . . 6 β’ (π¦ β (β΅βsuc π΄) β π¦ β On) |
3 | alephcard 10067 | . . . . . . . . 9 β’ (cardβ(β΅βsuc π΄)) = (β΅βsuc π΄) | |
4 | iscard 9972 | . . . . . . . . 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 485 | . . . . . . 7 β’ βπ¦ β (β΅βsuc π΄)π¦ βΊ (β΅βsuc π΄) |
7 | 6 | rspec 3241 | . . . . . 6 β’ (π¦ β (β΅βsuc π΄) β π¦ βΊ (β΅βsuc π΄)) |
8 | 2, 7 | jca 511 | . . . . 5 β’ (π¦ β (β΅βsuc π΄) β (π¦ β On β§ π¦ βΊ (β΅βsuc π΄))) |
9 | sdomel 9126 | . . . . . . 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 688 | . . . . . 6 β’ (π¦ β On β (π¦ βΊ (β΅βsuc π΄) β π¦ β (β΅βsuc π΄))) |
11 | 10 | imp 406 | . . . . 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 5144 | . . . . 5 β’ (π₯ = π¦ β (π₯ βΊ (β΅βsuc π΄) β π¦ βΊ (β΅βsuc π΄))) | |
14 | 13 | elrab 3678 | . . . 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 2723 | . 2 β’ (β΅βsuc π΄) = {π₯ β On β£ π₯ βΊ (β΅βsuc π΄)} |
17 | alephsucdom 10076 | . . 3 β’ (π΄ β On β (π₯ βΌ (β΅βπ΄) β π₯ βΊ (β΅βsuc π΄))) | |
18 | 17 | rabbidv 3434 | . 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 2785 | 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 395 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 class class class wbr 5141 Oncon0 6358 suc csuc 6360 βcfv 6537 βΌ cdom 8939 βΊ csdm 8940 cardccrd 9932 β΅cale 9933 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-oi 9507 df-har 9554 df-card 9936 df-aleph 9937 |
This theorem is referenced by: alephsuc3 10577 |
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