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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > r1rankcld | Structured version Visualization version GIF version |
Description: Any rank of the cumulative hierarchy is closed under the rank function. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
r1rankcld.1 | β’ (π β π΄ β (π 1βπ )) |
Ref | Expression |
---|---|
r1rankcld | β’ (π β (rankβπ΄) β (π 1βπ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onssr1 9864 | . . . 4 β’ (π β dom π 1 β π β (π 1βπ )) | |
2 | 1 | adantl 480 | . . 3 β’ ((π β§ π β dom π 1) β π β (π 1βπ )) |
3 | r1rankcld.1 | . . . . 5 β’ (π β π΄ β (π 1βπ )) | |
4 | rankr1ai 9831 | . . . . 5 β’ (π΄ β (π 1βπ ) β (rankβπ΄) β π ) | |
5 | 3, 4 | syl 17 | . . . 4 β’ (π β (rankβπ΄) β π ) |
6 | 5 | adantr 479 | . . 3 β’ ((π β§ π β dom π 1) β (rankβπ΄) β π ) |
7 | 2, 6 | sseldd 3983 | . 2 β’ ((π β§ π β dom π 1) β (rankβπ΄) β (π 1βπ )) |
8 | 3 | adantr 479 | . . 3 β’ ((π β§ Β¬ π β dom π 1) β π΄ β (π 1βπ )) |
9 | noel 4334 | . . . . . 6 β’ Β¬ π΄ β β | |
10 | 9 | a1i 11 | . . . . 5 β’ (Β¬ π β dom π 1 β Β¬ π΄ β β ) |
11 | ndmfv 6937 | . . . . 5 β’ (Β¬ π β dom π 1 β (π 1βπ ) = β ) | |
12 | 10, 11 | neleqtrrd 2852 | . . . 4 β’ (Β¬ π β dom π 1 β Β¬ π΄ β (π 1βπ )) |
13 | 12 | adantl 480 | . . 3 β’ ((π β§ Β¬ π β dom π 1) β Β¬ π΄ β (π 1βπ )) |
14 | 8, 13 | pm2.21dd 194 | . 2 β’ ((π β§ Β¬ π β dom π 1) β (rankβπ΄) β (π 1βπ )) |
15 | 7, 14 | pm2.61dan 811 | 1 β’ (π β (rankβπ΄) β (π 1βπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β wcel 2098 β wss 3949 β c0 4326 dom cdm 5682 βcfv 6553 π 1cr1 9795 rankcrnk 9796 |
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-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
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-int 4954 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-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-r1 9797 df-rank 9798 |
This theorem is referenced by: grurankcld 43719 |
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