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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 43580 | . . . 4 β’ Scott π΄ β V | |
2 | 1 | rankval4 9864 | . . 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 480 | . . . . . 6 β’ ((π β§ π₯ β Scott π΄) β π΅ β Scott π΄) |
6 | simpr 484 | . . . . . 6 β’ ((π β§ π₯ β Scott π΄) β π₯ β Scott π΄) | |
7 | 5, 6 | scottelrankd 43582 | . . . . 5 β’ ((π β§ π₯ β Scott π΄) β (rankβπ΅) β (rankβπ₯)) |
8 | 6, 5 | scottelrankd 43582 | . . . . 5 β’ ((π β§ π₯ β Scott π΄) β (rankβπ₯) β (rankβπ΅)) |
9 | 7, 8 | eqssd 3994 | . . . 4 β’ ((π β§ π₯ β Scott π΄) β (rankβπ΅) = (rankβπ₯)) |
10 | 9 | suceqd 43539 | . . 3 β’ ((π β§ π₯ β Scott π΄) β suc (rankβπ΅) = suc (rankβπ₯)) |
11 | 10 | iuneq2dv 5014 | . 2 β’ (π β βͺ π₯ β Scott π΄ suc (rankβπ΅) = βͺ π₯ β Scott π΄ suc (rankβπ₯)) |
12 | 4 | ne0d 4330 | . . 3 β’ (π β Scott π΄ β β ) |
13 | iunconst 4999 | . . 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 2772 | 1 β’ (π β (rankβScott π΄) = suc (rankβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β wne 2934 β c0 4317 βͺ ciun 4990 suc csuc 6360 βcfv 6537 rankcrnk 9760 Scott cscott 43570 |
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-reg 9589 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-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-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-ov 7408 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-r1 9761 df-rank 9762 df-scott 43571 |
This theorem is referenced by: gruscottcld 43584 |
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