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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33664 | . 2 ⊢ 0s = (∅ |s ∅) | |
2 | 0elpw 5223 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
3 | nulssgt 33638 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
5 | scutcl 33642 | . . 3 ⊢ (∅ <<s ∅ → (∅ |s ∅) ∈ No ) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (∅ |s ∅) ∈ No |
7 | 1, 6 | eqeltri 2830 | 1 ⊢ 0s ∈ No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ∅c0 4212 𝒫 cpw 4489 class class class wbr 5031 (class class class)co 7173 No csur 33489 <<s csslt 33621 |s cscut 33623 0s c0s 33662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-ord 6176 df-on 6177 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1o 8134 df-2o 8135 df-no 33492 df-slt 33493 df-bday 33494 df-sslt 33622 df-scut 33624 df-0s 33664 |
This theorem is referenced by: 1sno 33667 0slt1s 33669 bday1s 33671 made0 33703 negs0s 33770 addsid1 33773 |
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