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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 33665 | . . . . 5 ⊢ 0s ∈ No | |
2 | slerflex 33611 | . . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 0s ≤s 0s |
4 | 1 | elexi 3417 | . . . . 5 ⊢ 0s ∈ V |
5 | breq2 5034 | . . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s )) | |
6 | 4, 5 | rexsn 4573 | . . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s ) |
7 | 3, 6 | mpbir 234 | . . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥 |
8 | 7 | orci 864 | . 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ) |
9 | 0elpw 5222 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
10 | nulssgt 33637 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
12 | snssi 4696 | . . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ { 0s } ⊆ No |
14 | snex 5298 | . . . . . 6 ⊢ { 0s } ∈ V | |
15 | 14 | elpw 4492 | . . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) |
16 | 13, 15 | mpbir 234 | . . . 4 ⊢ { 0s } ∈ 𝒫 No |
17 | nulssgt 33637 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ |
19 | df-0s 33663 | . . 3 ⊢ 0s = (∅ |s ∅) | |
20 | df-1s 33664 | . . 3 ⊢ 1s = ({ 0s } |s ∅) | |
21 | sltrec 33659 | . . 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 234 | 1 ⊢ 0s <s 1s |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 {csn 4516 class class class wbr 5030 (class class class)co 7172 No csur 33488 <s cslt 33489 ≤s csle 33592 <<s csslt 33620 |s cscut 33622 0s c0s 33661 1s c1s 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 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-1o 8133 df-2o 8134 df-no 33491 df-slt 33492 df-bday 33493 df-sle 33593 df-sslt 33621 df-scut 33623 df-0s 33663 df-1s 33664 |
This theorem is referenced by: (None) |
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