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Mirrors > Home > MPE Home > Th. List > Mathboxes > oldssmade | Structured version Visualization version GIF version |
Description: The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.) |
Ref | Expression |
---|---|
oldssmade | β’ ( O βπ΄) β ( M βπ΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elold 34098 | . . . 4 β’ (π΄ β On β (π₯ β ( O βπ΄) β βπ β π΄ π₯ β ( M βπ))) | |
2 | onelss 6323 | . . . . . . . 8 β’ (π΄ β On β (π β π΄ β π β π΄)) | |
3 | 2 | imp 408 | . . . . . . 7 β’ ((π΄ β On β§ π β π΄) β π β π΄) |
4 | madess 34104 | . . . . . . 7 β’ ((π΄ β On β§ π β π΄) β ( M βπ) β ( M βπ΄)) | |
5 | 3, 4 | syldan 592 | . . . . . 6 β’ ((π΄ β On β§ π β π΄) β ( M βπ) β ( M βπ΄)) |
6 | 5 | sseld 3925 | . . . . 5 β’ ((π΄ β On β§ π β π΄) β (π₯ β ( M βπ) β π₯ β ( M βπ΄))) |
7 | 6 | rexlimdva 3149 | . . . 4 β’ (π΄ β On β (βπ β π΄ π₯ β ( M βπ) β π₯ β ( M βπ΄))) |
8 | 1, 7 | sylbid 239 | . . 3 β’ (π΄ β On β (π₯ β ( O βπ΄) β π₯ β ( M βπ΄))) |
9 | 8 | ssrdv 3932 | . 2 β’ (π΄ β On β ( O βπ΄) β ( M βπ΄)) |
10 | oldf 34086 | . . . . . 6 β’ O :OnβΆπ« No | |
11 | 10 | fdmi 6642 | . . . . 5 β’ dom O = On |
12 | 11 | eleq2i 2828 | . . . 4 β’ (π΄ β dom O β π΄ β On) |
13 | ndmfv 6836 | . . . 4 β’ (Β¬ π΄ β dom O β ( O βπ΄) = β ) | |
14 | 12, 13 | sylnbir 331 | . . 3 β’ (Β¬ π΄ β On β ( O βπ΄) = β ) |
15 | 0ss 4336 | . . 3 β’ β β ( M βπ΄) | |
16 | 14, 15 | eqsstrdi 3980 | . 2 β’ (Β¬ π΄ β On β ( O βπ΄) β ( M βπ΄)) |
17 | 9, 16 | pm2.61i 182 | 1 β’ ( O βπ΄) β ( M βπ΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 397 = wceq 1539 β wcel 2104 βwrex 3071 β wss 3892 β c0 4262 π« cpw 4539 dom cdm 5600 Oncon0 6281 βcfv 6458 No csur 33888 M cmade 34071 O cold 34072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-1o 8328 df-2o 8329 df-no 33891 df-slt 33892 df-bday 33893 df-sslt 34021 df-scut 34023 df-made 34076 df-old 34077 |
This theorem is referenced by: madeun 34111 madeoldsuc 34112 oldlim 34114 |
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