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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > scottrankd | Structured version Visualization version GIF version |
Description: Rank of a nonempty Scott's trick set. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
scottrankd.1 | β’ (π β π΅ β Scott π΄) |
Ref | Expression |
---|---|
scottrankd | β’ (π β (rankβScott π΄) = suc (rankβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scottex2 42617 | . . . 4 β’ Scott π΄ β V | |
2 | 1 | rankval4 9811 | . . 3 β’ (rankβScott π΄) = βͺ π₯ β Scott π΄ suc (rankβπ₯) |
3 | 2 | a1i 11 | . 2 β’ (π β (rankβScott π΄) = βͺ π₯ β Scott π΄ suc (rankβπ₯)) |
4 | scottrankd.1 | . . . . . . 7 β’ (π β π΅ β Scott π΄) | |
5 | 4 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β Scott π΄) β π΅ β Scott π΄) |
6 | simpr 486 | . . . . . 6 β’ ((π β§ π₯ β Scott π΄) β π₯ β Scott π΄) | |
7 | 5, 6 | scottelrankd 42619 | . . . . 5 β’ ((π β§ π₯ β Scott π΄) β (rankβπ΅) β (rankβπ₯)) |
8 | 6, 5 | scottelrankd 42619 | . . . . 5 β’ ((π β§ π₯ β Scott π΄) β (rankβπ₯) β (rankβπ΅)) |
9 | 7, 8 | eqssd 3965 | . . . 4 β’ ((π β§ π₯ β Scott π΄) β (rankβπ΅) = (rankβπ₯)) |
10 | 9 | suceqd 42576 | . . 3 β’ ((π β§ π₯ β Scott π΄) β suc (rankβπ΅) = suc (rankβπ₯)) |
11 | 10 | iuneq2dv 4982 | . 2 β’ (π β βͺ π₯ β Scott π΄ suc (rankβπ΅) = βͺ π₯ β Scott π΄ suc (rankβπ₯)) |
12 | 4 | ne0d 4299 | . . 3 β’ (π β Scott π΄ β β ) |
13 | iunconst 4967 | . . 3 β’ (Scott π΄ β β β βͺ π₯ β Scott π΄ suc (rankβπ΅) = suc (rankβπ΅)) | |
14 | 12, 13 | syl 17 | . 2 β’ (π β βͺ π₯ β Scott π΄ suc (rankβπ΅) = suc (rankβπ΅)) |
15 | 3, 11, 14 | 3eqtr2d 2779 | 1 β’ (π β (rankβScott π΄) = suc (rankβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 β c0 4286 βͺ ciun 4958 suc csuc 6323 βcfv 6500 rankcrnk 9707 Scott cscott 42607 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-reg 9536 ax-inf2 9585 |
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 df-scott 42608 |
This theorem is referenced by: gruscottcld 42621 |
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