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Mirrors > Home > MPE Home > Th. List > 0sno | Structured version Visualization version GIF version |
Description: Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.) |
Ref | Expression |
---|---|
0sno | ⊢ 0s ∈ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0s 27670 | . 2 ⊢ 0s = (∅ |s ∅) | |
2 | 0elpw 5354 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
3 | nulssgt 27644 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
5 | scutcl 27648 | . . 3 ⊢ (∅ <<s ∅ → (∅ |s ∅) ∈ No ) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (∅ |s ∅) ∈ No |
7 | 1, 6 | eqeltri 2828 | 1 ⊢ 0s ∈ No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∅c0 4322 𝒫 cpw 4602 class class class wbr 5148 (class class class)co 7412 No csur 27486 <<s csslt 27626 |s cscut 27628 0s c0s 27668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1o 8472 df-2o 8473 df-no 27489 df-slt 27490 df-bday 27491 df-sslt 27627 df-scut 27629 df-0s 27670 |
This theorem is referenced by: 1sno 27673 0slt1s 27675 bday1s 27677 cuteq0 27678 cuteq1 27679 sgt0ne0 27680 made0 27713 right1s 27735 0elold 27748 addsrid 27794 addslid 27798 addsproplem2 27800 addsfo 27813 sltaddpos1d 27841 sltaddpos2d 27842 addsgt0d 27844 negs0s 27852 negsproplem2 27854 negsproplem6 27858 negscl 27861 negsid 27866 negsdi 27875 slt0neg2d 27876 negsval2 27887 subsid1 27889 posdifsd 27917 sltsubposd 27918 muls01 27925 mulsrid 27926 mulsproplem2 27930 mulsproplem3 27931 mulsproplem4 27932 mulsproplem5 27933 mulsproplem6 27934 mulsproplem7 27935 mulsproplem8 27936 mulscl 27947 sltmul 27949 slemuld 27951 muls02 27954 mulsgt0 27957 mulsge0d 27959 sltmulneg1d 27989 mulscan2d 27992 slemul1ad 27995 sltmul12ad 27996 muls0ord 27998 precsexlem8 28025 precsexlem9 28026 precsexlem11 28028 recsex 28030 abs0s 28049 abssnid 28050 absmuls 28051 abssge0 28052 abssneg 28054 sleabs 28055 0ons 28062 n0ssno 28076 0n0s 28083 dfn0s2 28085 n0scut 28087 n0sge0 28090 nnsgt0 28091 elnns2 28093 recut 28104 0reno 28105 |
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