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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 | ⊢ 0s <s 1s |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0sno 34020 | . . . . 5 ⊢ 0s ∈ No | |
2 | slerflex 33966 | . . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 0s ≤s 0s |
4 | 1 | elexi 3451 | . . . . 5 ⊢ 0s ∈ V |
5 | breq2 5078 | . . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s )) | |
6 | 4, 5 | rexsn 4618 | . . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s ) |
7 | 3, 6 | mpbir 230 | . . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥 |
8 | 7 | orci 862 | . 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ) |
9 | 0elpw 5278 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
10 | nulssgt 33992 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
12 | snssi 4741 | . . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ { 0s } ⊆ No |
14 | snex 5354 | . . . . . 6 ⊢ { 0s } ∈ V | |
15 | 14 | elpw 4537 | . . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) |
16 | 13, 15 | mpbir 230 | . . . 4 ⊢ { 0s } ∈ 𝒫 No |
17 | nulssgt 33992 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ |
19 | df-0s 34018 | . . 3 ⊢ 0s = (∅ |s ∅) | |
20 | df-1s 34019 | . . 3 ⊢ 1s = ({ 0s } |s ∅) | |
21 | sltrec 34014 | . . 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 690 | . 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )) |
23 | 8, 22 | mpbir 230 | 1 ⊢ 0s <s 1s |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 844 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 class class class wbr 5074 (class class class)co 7275 No csur 33843 <s cslt 33844 ≤s csle 33947 <<s csslt 33975 |s cscut 33977 0s c0s 34016 1s c1s 34017 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1o 8297 df-2o 8298 df-no 33846 df-slt 33847 df-bday 33848 df-sle 33948 df-sslt 33976 df-scut 33978 df-0s 34018 df-1s 34019 |
This theorem is referenced by: (None) |
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