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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 27187 | . . . . 5 âĒ 0s â No | |
2 | slerflex 27127 | . . . . 5 âĒ ( 0s â No â 0s âĪs 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 âĒ 0s âĪs 0s |
4 | 1 | elexi 3463 | . . . . 5 âĒ 0s â V |
5 | breq2 5110 | . . . . 5 âĒ (ðĨ = 0s â ( 0s âĪs ðĨ â 0s âĪs 0s )) | |
6 | 4, 5 | rexsn 4644 | . . . 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 864 | . 2 âĒ (âðĨ â { 0s } 0s âĪs ðĨ âĻ âðĶ â â ðĶ âĪs 1s ) |
9 | 0elpw 5312 | . . . 4 âĒ â â ðŦ No | |
10 | nulssgt 27159 | . . . 4 âĒ (â â ðŦ No â â <<s â ) | |
11 | 9, 10 | ax-mp 5 | . . 3 âĒ â <<s â |
12 | snssi 4769 | . . . . . 6 âĒ ( 0s â No â { 0s } â No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 âĒ { 0s } â No |
14 | snex 5389 | . . . . . 6 âĒ { 0s } â V | |
15 | 14 | elpw 4565 | . . . . 5 âĒ ({ 0s } â ðŦ No â { 0s } â No ) |
16 | 13, 15 | mpbir 230 | . . . 4 âĒ { 0s } â ðŦ No |
17 | nulssgt 27159 | . . . 4 âĒ ({ 0s } â ðŦ No â { 0s } <<s â ) | |
18 | 16, 17 | ax-mp 5 | . . 3 âĒ { 0s } <<s â |
19 | df-0s 27185 | . . 3 âĒ 0s = (â |s â ) | |
20 | df-1s 27186 | . . 3 âĒ 1s = ({ 0s } |s â ) | |
21 | sltrec 27181 | . . 3 âĒ (((â <<s â ⧠{ 0s } <<s â ) ⧠( 0s = (â |s 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 692 | . 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 846 = wceq 1542 â wcel 2107 âwrex 3070 â wss 3911 â c0 4283 ðŦ cpw 4561 {csn 4587 class class class wbr 5106 (class class class)co 7358 No csur 27004 <s cslt 27005 âĪs csle 27108 <<s csslt 27142 |s cscut 27144 0s c0s 27183 1s c1s 27184 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-bday 27009 df-sle 27109 df-sslt 27143 df-scut 27145 df-0s 27185 df-1s 27186 |
This theorem is referenced by: left1s 27246 right1s 27247 |
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