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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 5393 | . . 3 β’ {π΄} β V | |
2 | 1 | a1i 11 | . 2 β’ (π΄ β No β {π΄} β V) |
3 | fvexd 6862 | . 2 β’ (π΄ β No β ( R βπ΄) β V) | |
4 | snssi 4773 | . 2 β’ (π΄ β No β {π΄} β No ) | |
5 | rightf 27218 | . . . 4 β’ R : No βΆπ« No | |
6 | 5 | ffvelcdmi 7039 | . . 3 β’ (π΄ β No β ( R βπ΄) β π« No ) |
7 | 6 | elpwid 4574 | . 2 β’ (π΄ β No β ( R βπ΄) β No ) |
8 | velsn 4607 | . . . 4 β’ (π₯ β {π΄} β π₯ = π΄) | |
9 | rightval 27216 | . . . . . . . . . 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 2824 | . . . . . . . 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 3430 | . . . . . . . 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 287 | . . . . . . 7 β’ (π΄ β No β (π¦ β ( R βπ΄) β (π¦ β ( O β( bday βπ΄)) β§ π΄ <s π¦))) |
14 | 13 | simplbda 501 | . . . . . 6 β’ ((π΄ β No β§ π¦ β ( R βπ΄)) β π΄ <s π¦) |
15 | breq1 5113 | . . . . . 6 β’ (π₯ = π΄ β (π₯ <s π¦ β π΄ <s π¦)) | |
16 | 14, 15 | syl5ibr 246 | . . . . 5 β’ (π₯ = π΄ β ((π΄ β No β§ π¦ β ( R βπ΄)) β π₯ <s π¦)) |
17 | 16 | expd 417 | . . . 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 1115 | . 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 27153 | 1 β’ (π΄ β No β {π΄} <<s ( R βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3410 Vcvv 3448 π« cpw 4565 {csn 4591 class class class wbr 5110 βcfv 6501 No csur 27004 <s cslt 27005 bday cbday 27006 <<s csslt 27142 O cold 27195 R cright 27198 |
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 df-right 27203 |
This theorem is referenced by: lltropt 27224 madebdaylemlrcut 27250 mulsproplem6 27406 mulsproplem7 27407 mulsproplem8 27408 mulsproplem9 27409 |
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