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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 33614 | . . . . 5 ⊢ 0s ∈ No | |
2 | slerflex 33563 | . . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 0s ≤s 0s |
4 | 1 | elexi 3429 | . . . . 5 ⊢ 0s ∈ V |
5 | breq2 5040 | . . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s )) | |
6 | 4, 5 | rexsn 4580 | . . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s ) |
7 | 3, 6 | mpbir 234 | . . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥 |
8 | 7 | orci 862 | . 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ) |
9 | 0elpw 5228 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
10 | nulssgt 33587 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
12 | snssi 4701 | . . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ { 0s } ⊆ No |
14 | snex 5304 | . . . . . 6 ⊢ { 0s } ∈ V | |
15 | 14 | elpw 4501 | . . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) |
16 | 13, 15 | mpbir 234 | . . . 4 ⊢ { 0s } ∈ 𝒫 No |
17 | nulssgt 33587 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ |
19 | df-0s 33612 | . . 3 ⊢ 0s = (∅ |s ∅) | |
20 | df-1s 33613 | . . 3 ⊢ 1s = ({ 0s } |s ∅) | |
21 | sltrec 33609 | . . 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 692 | . 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )) |
23 | 8, 22 | mpbir 234 | 1 ⊢ 0s <s 1s |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 ⊆ wss 3860 ∅c0 4227 𝒫 cpw 4497 {csn 4525 class class class wbr 5036 (class class class)co 7156 No csur 33440 <s cslt 33441 ≤s csle 33544 <<s csslt 33572 |s cscut 33574 0s c0s 33610 1s c1s 33611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 df-bday 33445 df-sle 33545 df-sslt 33573 df-scut 33575 df-0s 33612 df-1s 33613 |
This theorem is referenced by: (None) |
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