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Mirrors > Home > MPE Home > Th. List > oldbdayim | Structured version Visualization version GIF version |
Description: If π is in the old set for π΄, then the birthday of π is less than π΄. (Contributed by Scott Fenton, 10-Aug-2024.) |
Ref | Expression |
---|---|
oldbdayim | β’ (π β ( O βπ΄) β ( bday βπ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6884 | . . 3 β’ (π β ( O βπ΄) β π΄ β dom O ) | |
2 | oldf 27209 | . . . 4 β’ O :OnβΆπ« No | |
3 | 2 | fdmi 6685 | . . 3 β’ dom O = On |
4 | 1, 3 | eleqtrdi 2848 | . 2 β’ (π β ( O βπ΄) β π΄ β On) |
5 | elold 27221 | . . 3 β’ (π΄ β On β (π β ( O βπ΄) β βπ β π΄ π β ( M βπ))) | |
6 | madebdayim 27239 | . . . . . 6 β’ (π β ( M βπ) β ( bday βπ) β π) | |
7 | 6 | ad2antll 728 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π) |
8 | simprl 770 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β π β π΄) | |
9 | bdayelon 27138 | . . . . . . 7 β’ ( bday βπ) β On | |
10 | ontr2 6369 | . . . . . . 7 β’ ((( bday βπ) β On β§ π΄ β On) β ((( bday βπ) β π β§ π β π΄) β ( bday βπ) β π΄)) | |
11 | 9, 10 | mpan 689 | . . . . . 6 β’ (π΄ β On β ((( bday βπ) β π β§ π β π΄) β ( bday βπ) β π΄)) |
12 | 11 | adantr 482 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ((( bday βπ) β π β§ π β π΄) β ( bday βπ) β π΄)) |
13 | 7, 8, 12 | mp2and 698 | . . . 4 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π΄) |
14 | 13 | rexlimdvaa 3154 | . . 3 β’ (π΄ β On β (βπ β π΄ π β ( M βπ) β ( bday βπ) β π΄)) |
15 | 5, 14 | sylbid 239 | . 2 β’ (π΄ β On β (π β ( O βπ΄) β ( bday βπ) β π΄)) |
16 | 4, 15 | mpcom 38 | 1 β’ (π β ( O βπ΄) β ( bday βπ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 βwrex 3074 β wss 3915 π« cpw 4565 dom cdm 5638 Oncon0 6322 βcfv 6501 No csur 27004 bday cbday 27006 M cmade 27194 O cold 27195 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-1o 8417 df-2o 8418 df-no 27007 df-slt 27008 df-bday 27009 df-sslt 27143 df-scut 27145 df-made 27199 df-old 27200 |
This theorem is referenced by: oldirr 27241 oldbday 27252 negsproplem2 27349 mulsproplem3 27403 mulsproplem4 27404 mulsproplem5 27405 mulsproplem6 27406 mulsproplem7 27407 mulsproplem8 27408 mulsproplem9 27409 |
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