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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 9775 | . . . 4 β’ (π β dom π 1 β π β (π 1βπ )) | |
2 | 1 | adantl 483 | . . 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 9742 | . . . . 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 482 | . . 3 β’ ((π β§ π β dom π 1) β (rankβπ΄) β π ) |
7 | 2, 6 | sseldd 3949 | . 2 β’ ((π β§ π β dom π 1) β (rankβπ΄) β (π 1βπ )) |
8 | 3 | adantr 482 | . . 3 β’ ((π β§ Β¬ π β dom π 1) β π΄ β (π 1βπ )) |
9 | noel 4294 | . . . . . 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 6881 | . . . . 5 β’ (Β¬ π β dom π 1 β (π 1βπ ) = β ) | |
12 | 10, 11 | neleqtrrd 2857 | . . . 4 β’ (Β¬ π β dom π 1 β Β¬ π΄ β (π 1βπ )) |
13 | 12 | adantl 483 | . . 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 812 | 1 β’ (π β (rankβπ΄) β (π 1βπ )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β wcel 2107 β wss 3914 β c0 4286 dom cdm 5637 βcfv 6500 π 1cr1 9706 rankcrnk 9707 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-r1 9708 df-rank 9709 |
This theorem is referenced by: grurankcld 42605 |
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