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Mirrors > Home > MPE Home > Th. List > 0slt1s | Structured version Visualization version GIF version |
Description: Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
0slt1s | âĒ 0s <s 1s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 27710 | . . . . 5 âĒ 0s â No | |
2 | slerflex 27647 | . . . . 5 âĒ ( 0s â No â 0s âĪs 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 âĒ 0s âĪs 0s |
4 | 1 | elexi 3488 | . . . . 5 âĒ 0s â V |
5 | breq2 5145 | . . . . 5 âĒ (ðĨ = 0s â ( 0s âĪs ðĨ â 0s âĪs 0s )) | |
6 | 4, 5 | rexsn 4681 | . . . 4 âĒ (âðĨ â { 0s } 0s âĪs ðĨ â 0s âĪs 0s ) |
7 | 3, 6 | mpbir 230 | . . 3 âĒ âðĨ â { 0s } 0s âĪs ðĨ |
8 | 7 | orci 862 | . 2 âĒ (âðĨ â { 0s } 0s âĪs ðĨ âĻ âðĶ â â ðĶ âĪs 1s ) |
9 | 0elpw 5347 | . . . 4 âĒ â â ðŦ No | |
10 | nulssgt 27682 | . . . 4 âĒ (â â ðŦ No â â <<s â ) | |
11 | 9, 10 | ax-mp 5 | . . 3 âĒ â <<s â |
12 | snssi 4806 | . . . . . 6 âĒ ( 0s â No â { 0s } â No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 âĒ { 0s } â No |
14 | snex 5424 | . . . . . 6 âĒ { 0s } â V | |
15 | 14 | elpw 4601 | . . . . 5 âĒ ({ 0s } â ðŦ No â { 0s } â No ) |
16 | 13, 15 | mpbir 230 | . . . 4 âĒ { 0s } â ðŦ No |
17 | nulssgt 27682 | . . . 4 âĒ ({ 0s } â ðŦ No â { 0s } <<s â ) | |
18 | 16, 17 | ax-mp 5 | . . 3 âĒ { 0s } <<s â |
19 | df-0s 27708 | . . 3 âĒ 0s = (â |s â ) | |
20 | df-1s 27709 | . . 3 âĒ 1s = ({ 0s } |s â ) | |
21 | sltrec 27704 | . . 3 âĒ (((â <<s â â§ { 0s } <<s â ) â§ ( 0s = (â |s â ) â§ 1s = ({ 0s } |s â ))) â ( 0s <s 1s â (âðĨ â { 0s } 0s âĪs ðĨ âĻ âðĶ â â ðĶ âĪs 1s ))) | |
22 | 11, 18, 19, 20, 21 | mp4an 690 | . 2 âĒ ( 0s <s 1s â (âðĨ â { 0s } 0s âĪs ðĨ âĻ âðĶ â â ðĶ âĪs 1s )) |
23 | 8, 22 | mpbir 230 | 1 âĒ 0s <s 1s |
Colors of variables: wff setvar class |
Syntax hints: â wb 205 âĻ wo 844 = wceq 1533 â wcel 2098 âwrex 3064 â wss 3943 â c0 4317 ðŦ cpw 4597 {csn 4623 class class class wbr 5141 (class class class)co 7404 No csur 27524 <s cslt 27525 âĪs csle 27628 <<s csslt 27664 |s cscut 27666 0s c0s 27706 1s c1s 27707 |
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-pr 5420 ax-un 7721 |
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-ord 6360 df-on 6361 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1o 8464 df-2o 8465 df-no 27527 df-slt 27528 df-bday 27529 df-sle 27629 df-sslt 27665 df-scut 27667 df-0s 27708 df-1s 27709 |
This theorem is referenced by: left1s 27772 right1s 27773 divs1 28054 precsexlem9 28064 om2noseqlt 28123 n0scut 28154 n0sge0 28157 1nns 28166 nnsrecgt0d 28170 0reno 28176 |
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