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Mirrors > Home > MPE Home > Th. List > ssltright | Structured version Visualization version GIF version |
Description: A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
ssltright | β’ (π΄ β No β {π΄} <<s ( R βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5427 | . . 3 β’ {π΄} β V | |
2 | 1 | a1i 11 | . 2 β’ (π΄ β No β {π΄} β V) |
3 | fvexd 6907 | . 2 β’ (π΄ β No β ( R βπ΄) β V) | |
4 | snssi 4807 | . 2 β’ (π΄ β No β {π΄} β No ) | |
5 | rightf 27811 | . . . 4 β’ R : No βΆπ« No | |
6 | 5 | ffvelcdmi 7088 | . . 3 β’ (π΄ β No β ( R βπ΄) β π« No ) |
7 | 6 | elpwid 4607 | . 2 β’ (π΄ β No β ( R βπ΄) β No ) |
8 | velsn 4640 | . . . 4 β’ (π₯ β {π΄} β π₯ = π΄) | |
9 | rightval 27809 | . . . . . . . . . 10 β’ ( R βπ΄) = {π¦ β ( O β( bday βπ΄)) β£ π΄ <s π¦} | |
10 | 9 | a1i 11 | . . . . . . . . 9 β’ (π΄ β No β ( R βπ΄) = {π¦ β ( O β( bday βπ΄)) β£ π΄ <s π¦}) |
11 | 10 | eleq2d 2811 | . . . . . . . 8 β’ (π΄ β No β (π¦ β ( R βπ΄) β π¦ β {π¦ β ( O β( bday βπ΄)) β£ π΄ <s π¦})) |
12 | rabid 3440 | . . . . . . . 8 β’ (π¦ β {π¦ β ( O β( bday βπ΄)) β£ π΄ <s π¦} β (π¦ β ( O β( bday βπ΄)) β§ π΄ <s π¦)) | |
13 | 11, 12 | bitrdi 286 | . . . . . . 7 β’ (π΄ β No β (π¦ β ( R βπ΄) β (π¦ β ( O β( bday βπ΄)) β§ π΄ <s π¦))) |
14 | 13 | simplbda 498 | . . . . . 6 β’ ((π΄ β No β§ π¦ β ( R βπ΄)) β π΄ <s π¦) |
15 | breq1 5146 | . . . . . 6 β’ (π₯ = π΄ β (π₯ <s π¦ β π΄ <s π¦)) | |
16 | 14, 15 | imbitrrid 245 | . . . . 5 β’ (π₯ = π΄ β ((π΄ β No β§ π¦ β ( R βπ΄)) β π₯ <s π¦)) |
17 | 16 | expd 414 | . . . 4 β’ (π₯ = π΄ β (π΄ β No β (π¦ β ( R βπ΄) β π₯ <s π¦))) |
18 | 8, 17 | sylbi 216 | . . 3 β’ (π₯ β {π΄} β (π΄ β No β (π¦ β ( R βπ΄) β π₯ <s π¦))) |
19 | 18 | 3imp21 1111 | . 2 β’ ((π΄ β No β§ π₯ β {π΄} β§ π¦ β ( R βπ΄)) β π₯ <s π¦) |
20 | 2, 3, 4, 7, 19 | ssltd 27742 | 1 β’ (π΄ β No β {π΄} <<s ( R βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3419 Vcvv 3463 π« cpw 4598 {csn 4624 class class class wbr 5143 βcfv 6543 No csur 27591 <s cslt 27592 bday cbday 27593 <<s csslt 27731 O cold 27788 R cright 27791 |
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 df-right 27796 |
This theorem is referenced by: lltropt 27817 madebdaylemlrcut 27843 mulsproplem5 28042 mulsproplem6 28043 mulsproplem7 28044 mulsproplem8 28045 mulsuniflem 28071 |
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