Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0slt1s Structured version   Visualization version   GIF version

Theorem 0slt1s 34023
Description: Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
Assertion
Ref Expression
0slt1s 0s <s 1s

Proof of Theorem 0slt1s
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0sno 34020 . . . . 5 0s ∈ No
2 slerflex 33966 . . . . 5 ( 0s ∈ No → 0s ≤s 0s )
31, 2ax-mp 5 . . . 4 0s ≤s 0s
41elexi 3451 . . . . 5 0s ∈ V
5 breq2 5078 . . . . 5 (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
64, 5rexsn 4618 . . . 4 (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s )
73, 6mpbir 230 . . 3 𝑥 ∈ { 0s } 0s ≤s 𝑥
87orci 862 . 2 (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )
9 0elpw 5278 . . . 4 ∅ ∈ 𝒫 No
10 nulssgt 33992 . . . 4 (∅ ∈ 𝒫 No → ∅ <<s ∅)
119, 10ax-mp 5 . . 3 ∅ <<s ∅
12 snssi 4741 . . . . . 6 ( 0s ∈ No → { 0s } ⊆ No )
131, 12ax-mp 5 . . . . 5 { 0s } ⊆ No
14 snex 5354 . . . . . 6 { 0s } ∈ V
1514elpw 4537 . . . . 5 ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No )
1613, 15mpbir 230 . . . 4 { 0s } ∈ 𝒫 No
17 nulssgt 33992 . . . 4 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
1816, 17ax-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 )))
2211, 18, 19, 20, 21mp4an 690 . 2 ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))
238, 22mpbir 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)
  Copyright terms: Public domain W3C validator