![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 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 27886 | . . . . 5 ⊢ 0s ∈ No | |
2 | slerflex 27823 | . . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ 0s ≤s 0s |
4 | 1 | elexi 3501 | . . . . 5 ⊢ 0s ∈ V |
5 | breq2 5152 | . . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s )) | |
6 | 4, 5 | rexsn 4687 | . . . 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 5362 | . . . 4 ⊢ ∅ ∈ 𝒫 No | |
10 | nulssgt 27858 | . . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ ∅ <<s ∅ |
12 | snssi 4813 | . . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No ) | |
13 | 1, 12 | ax-mp 5 | . . . . 5 ⊢ { 0s } ⊆ No |
14 | snex 5442 | . . . . . 6 ⊢ { 0s } ∈ V | |
15 | 14 | elpw 4609 | . . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No ) |
16 | 13, 15 | mpbir 231 | . . . 4 ⊢ { 0s } ∈ 𝒫 No |
17 | nulssgt 27858 | . . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ { 0s } <<s ∅ |
19 | df-0s 27884 | . . 3 ⊢ 0s = (∅ |s ∅) | |
20 | df-1s 27885 | . . 3 ⊢ 1s = ({ 0s } |s ∅) | |
21 | sltrec 27880 | . . 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 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 class class class wbr 5148 (class class class)co 7431 No csur 27699 <s cslt 27700 ≤s csle 27804 <<s csslt 27840 |s cscut 27842 0s c0s 27882 1s c1s 27883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 |
This theorem is referenced by: left1s 27948 right1s 27949 sltp1d 28063 divs1 28244 precsexlem9 28254 om2noseqlt 28320 n0scut 28353 n0sge0 28356 1nns 28367 nnsrecgt0d 28371 nohalf 28422 expsne0 28429 expsgt0 28430 cutpw2 28432 pw2bday 28433 0reno 28444 |
Copyright terms: Public domain | W3C validator |