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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 6922 | . . 3 β’ (π β ( O βπ΄) β π΄ β dom O ) | |
2 | oldf 27739 | . . . 4 β’ O :OnβΆπ« No | |
3 | 2 | fdmi 6723 | . . 3 β’ dom O = On |
4 | 1, 3 | eleqtrdi 2837 | . 2 β’ (π β ( O βπ΄) β π΄ β On) |
5 | elold 27751 | . . 3 β’ (π΄ β On β (π β ( O βπ΄) β βπ β π΄ π β ( M βπ))) | |
6 | madebdayim 27769 | . . . . . 6 β’ (π β ( M βπ) β ( bday βπ) β π) | |
7 | 6 | ad2antll 726 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π) |
8 | simprl 768 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β π β π΄) | |
9 | bdayelon 27664 | . . . . . . 7 β’ ( bday βπ) β On | |
10 | ontr2 6405 | . . . . . . 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 687 | . . . . . 6 β’ (π΄ β On β ((( bday βπ) β π β§ π β π΄) β ( bday βπ) β π΄)) |
12 | 11 | adantr 480 | . . . . 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 696 | . . . 4 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π΄) |
14 | 13 | rexlimdvaa 3150 | . . 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 395 β wcel 2098 βwrex 3064 β wss 3943 π« cpw 4597 dom cdm 5669 Oncon0 6358 βcfv 6537 No csur 27528 bday cbday 27530 M cmade 27724 O cold 27725 |
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 |
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-rmo 3370 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-tp 4628 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-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-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-1o 8467 df-2o 8468 df-no 27531 df-slt 27532 df-bday 27533 df-sslt 27669 df-scut 27671 df-made 27729 df-old 27730 |
This theorem is referenced by: oldirr 27771 oldbday 27782 negsproplem2 27896 negsbdaylem 27923 mulsproplem2 27972 mulsproplem3 27973 mulsproplem4 27974 mulsproplem5 27975 mulsproplem6 27976 mulsproplem7 27977 mulsproplem8 27978 |
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