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| Mirrors > Home > MPE Home > Th. List > 1sno | Structured version Visualization version GIF version | ||
| Description: Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) |
| Ref | Expression |
|---|---|
| 1sno | ⊢ 1s ∈ No |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-1s 27870 | . 2 ⊢ 1s = ({ 0s } |s ∅) | |
| 2 | 0sno 27871 | . . . . 5 ⊢ 0s ∈ No | |
| 3 | snelpwi 5448 | . . . . 5 ⊢ ( 0s ∈ No → { 0s } ∈ 𝒫 No ) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ { 0s } ∈ 𝒫 No |
| 5 | nulssgt 27843 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
| 6 | 4, 5 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ |
| 7 | scutcl 27847 | . . 3 ⊢ ({ 0s } <<s ∅ → ({ 0s } |s ∅) ∈ No ) | |
| 8 | 6, 7 | ax-mp 5 | . 2 ⊢ ({ 0s } |s ∅) ∈ No |
| 9 | 1, 8 | eqeltri 2837 | 1 ⊢ 1s ∈ No |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 (class class class)co 7431 No csur 27684 <<s csslt 27825 |s cscut 27827 0s c0s 27867 1s c1s 27868 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 |
| This theorem is referenced by: cuteq1 27878 right1s 27934 peano2no 28017 sltp1d 28048 negs1s 28059 sltm1d 28131 mulsrid 28139 mulslid 28168 divs1 28229 precsexlem8 28238 precsexlem9 28239 precsexlem10 28240 precsexlem11 28241 divsrecd 28258 divsdird 28259 1ons 28280 om2noseqlt 28305 n0scut 28338 n0ons 28339 n0sge0 28341 n0s0suc 28345 nnsge1 28346 n0addscl 28347 n0mulscl 28348 1n0s 28351 n0sbday 28354 nnsrecgt0d 28356 n0s0m1 28359 n0subs 28360 n0p1nns 28361 dfnns2 28362 nnsind 28363 nnzs 28372 0zs 28374 elzn0s 28384 peano5uzs 28390 zscut 28393 1p1e2s 28400 no2times 28401 n0seo 28405 zseo 28406 nohalf 28407 expsval 28408 exps1 28411 expsp1 28412 expscl 28413 cutpw2 28417 pw2bday 28418 addhalfcut 28419 pw2cut 28420 zs12bday 28424 recut 28428 0reno 28429 renegscl 28430 readdscl 28431 remulscllem1 28432 remulscl 28434 |
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