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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 6929 | . . 3 β’ (π β ( O βπ΄) β π΄ β dom O ) | |
2 | oldf 27802 | . . . 4 β’ O :OnβΆπ« No | |
3 | 2 | fdmi 6729 | . . 3 β’ dom O = On |
4 | 1, 3 | eleqtrdi 2835 | . 2 β’ (π β ( O βπ΄) β π΄ β On) |
5 | elold 27814 | . . 3 β’ (π΄ β On β (π β ( O βπ΄) β βπ β π΄ π β ( M βπ))) | |
6 | madebdayim 27832 | . . . . . 6 β’ (π β ( M βπ) β ( bday βπ) β π) | |
7 | 6 | ad2antll 727 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π) |
8 | simprl 769 | . . . . 5 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β π β π΄) | |
9 | bdayelon 27727 | . . . . . . 7 β’ ( bday βπ) β On | |
10 | ontr2 6411 | . . . . . . 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 688 | . . . . . 6 β’ (π΄ β On β ((( bday βπ) β π β§ π β π΄) β ( bday βπ) β π΄)) |
12 | 11 | adantr 479 | . . . . 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 697 | . . . 4 β’ ((π΄ β On β§ (π β π΄ β§ π β ( M βπ))) β ( bday βπ) β π΄) |
14 | 13 | rexlimdvaa 3146 | . . 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 394 β wcel 2098 βwrex 3060 β wss 3939 π« cpw 4598 dom cdm 5672 Oncon0 6364 βcfv 6543 No csur 27591 bday cbday 27593 M cmade 27787 O cold 27788 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-1o 8485 df-2o 8486 df-no 27594 df-slt 27595 df-bday 27596 df-sslt 27732 df-scut 27734 df-made 27792 df-old 27793 |
This theorem is referenced by: oldirr 27834 oldbday 27845 negsproplem2 27959 negsbdaylem 27986 mulsproplem2 28039 mulsproplem3 28040 mulsproplem4 28041 mulsproplem5 28042 mulsproplem6 28043 mulsproplem7 28044 mulsproplem8 28045 |
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