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Mirrors > Home > MPE Home > Th. List > 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 27361 | . . . 4 β’ (π΄ β On β (π₯ β ( O βπ΄) β βπ β π΄ π₯ β ( M βπ))) | |
2 | onelss 6406 | . . . . . . . 8 β’ (π΄ β On β (π β π΄ β π β π΄)) | |
3 | 2 | imp 407 | . . . . . . 7 β’ ((π΄ β On β§ π β π΄) β π β π΄) |
4 | madess 27368 | . . . . . . 7 β’ ((π΄ β On β§ π β π΄) β ( M βπ) β ( M βπ΄)) | |
5 | 3, 4 | syldan 591 | . . . . . 6 β’ ((π΄ β On β§ π β π΄) β ( M βπ) β ( M βπ΄)) |
6 | 5 | sseld 3981 | . . . . 5 β’ ((π΄ β On β§ π β π΄) β (π₯ β ( M βπ) β π₯ β ( M βπ΄))) |
7 | 6 | rexlimdva 3155 | . . . 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 3988 | . 2 β’ (π΄ β On β ( O βπ΄) β ( M βπ΄)) |
10 | oldf 27349 | . . . . . 6 β’ O :OnβΆπ« No | |
11 | 10 | fdmi 6729 | . . . . 5 β’ dom O = On |
12 | 11 | eleq2i 2825 | . . . 4 β’ (π΄ β dom O β π΄ β On) |
13 | ndmfv 6926 | . . . 4 β’ (Β¬ π΄ β dom O β ( O βπ΄) = β ) | |
14 | 12, 13 | sylnbir 330 | . . 3 β’ (Β¬ π΄ β On β ( O βπ΄) = β ) |
15 | 0ss 4396 | . . 3 β’ β β ( M βπ΄) | |
16 | 14, 15 | eqsstrdi 4036 | . 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 396 = wceq 1541 β wcel 2106 βwrex 3070 β wss 3948 β c0 4322 π« cpw 4602 dom cdm 5676 Oncon0 6364 βcfv 6543 No csur 27140 M cmade 27334 O cold 27335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-1o 8465 df-2o 8466 df-no 27143 df-slt 27144 df-bday 27145 df-sslt 27280 df-scut 27282 df-made 27339 df-old 27340 |
This theorem is referenced by: madeun 27375 madeoldsuc 27376 oldlim 27378 |
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