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| Description: Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| 0slt1s | ⊢ 0s <s 1s | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0sno 27872 | . . . . 5 ⊢ 0s ∈ No | |
| 2 | slerflex 27809 | . . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
| 3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 0s ≤s 0s | 
| 4 | 1 | elexi 3502 | . . . . 5 ⊢ 0s ∈ V | 
| 5 | breq2 5146 | . . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s )) | |
| 6 | 4, 5 | rexsn 4681 | . . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s ) | 
| 7 | 3, 6 | mpbir 231 | . . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥 | 
| 8 | 7 | orci 865 | . 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ) | 
| 9 | 0elpw 5355 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
| 10 | nulssgt 27844 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
| 11 | 9, 10 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ | 
| 12 | snssi 4807 | . . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No ) | |
| 13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ { 0s } ⊆ No | 
| 14 | snex 5435 | . . . . . 6 ⊢ { 0s } ∈ V | |
| 15 | 14 | elpw 4603 | . . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) | 
| 16 | 13, 15 | mpbir 231 | . . . 4 ⊢ { 0s } ∈ 𝒫 No | 
| 17 | nulssgt 27844 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
| 18 | 16, 17 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ | 
| 19 | df-0s 27870 | . . 3 ⊢ 0s = (∅ |s ∅) | |
| 20 | df-1s 27871 | . . 3 ⊢ 1s = ({ 0s } |s ∅) | |
| 21 | sltrec 27866 | . . 3 ⊢ (((∅ <<s ∅ ∧ { 0s } <<s ∅) ∧ ( 0s = (∅ |s ∅) ∧ 1s = ({ 0s } |s ∅))) → ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))) | |
| 22 | 11, 18, 19, 20, 21 | mp4an 693 | . 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )) | 
| 23 | 8, 22 | mpbir 231 | 1 ⊢ 0s <s 1s | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 ∅c0 4332 𝒫 cpw 4599 {csn 4625 class class class wbr 5142 (class class class)co 7432 No csur 27685 <s cslt 27686 ≤s csle 27790 <<s csslt 27826 |s cscut 27828 0s c0s 27868 1s c1s 27869 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-bday 27690 df-sle 27791 df-sslt 27827 df-scut 27829 df-0s 27870 df-1s 27871 | 
| This theorem is referenced by: left1s 27934 right1s 27935 sltp1d 28049 divs1 28230 precsexlem9 28240 om2noseqlt 28306 n0scut 28339 n0sge0 28342 1nns 28353 nnsrecgt0d 28357 nohalf 28408 expsne0 28415 expsgt0 28416 cutpw2 28418 pw2bday 28419 0reno 28430 | 
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