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Theorem 0slt1s 27713
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 ð‘Ĩ ð‘Ķ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0sno 27710 . . . . 5 0s ∈ No
2 slerflex 27647 . . . . 5 ( 0s ∈ No → 0s â‰Īs 0s )
31, 2ax-mp 5 . . . 4 0s â‰Īs 0s
41elexi 3488 . . . . 5 0s ∈ V
5 breq2 5145 . . . . 5 (ð‘Ĩ = 0s → ( 0s â‰Īs ð‘Ĩ ↔ 0s â‰Īs 0s ))
64, 5rexsn 4681 . . . 4 (∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ ↔ 0s â‰Īs 0s )
73, 6mpbir 230 . . 3 ∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ
87orci 862 . 2 (∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ âˆĻ ∃ð‘Ķ ∈ ∅ ð‘Ķ â‰Īs 1s )
9 0elpw 5347 . . . 4 ∅ ∈ ð’Ŧ No
10 nulssgt 27682 . . . 4 (∅ ∈ ð’Ŧ No → ∅ <<s ∅)
119, 10ax-mp 5 . . 3 ∅ <<s ∅
12 snssi 4806 . . . . . 6 ( 0s ∈ No → { 0s } ⊆ No )
131, 12ax-mp 5 . . . . 5 { 0s } ⊆ No
14 snex 5424 . . . . . 6 { 0s } ∈ V
1514elpw 4601 . . . . 5 ({ 0s } ∈ ð’Ŧ No ↔ { 0s } ⊆ No )
1613, 15mpbir 230 . . . 4 { 0s } ∈ ð’Ŧ No
17 nulssgt 27682 . . . 4 ({ 0s } ∈ ð’Ŧ No → { 0s } <<s ∅)
1816, 17ax-mp 5 . . 3 { 0s } <<s ∅
19 df-0s 27708 . . 3 0s = (∅ |s ∅)
20 df-1s 27709 . . 3 1s = ({ 0s } |s ∅)
21 sltrec 27704 . . 3 (((∅ <<s ∅ ∧ { 0s } <<s ∅) ∧ ( 0s = (∅ |s ∅) ∧ 1s = ({ 0s } |s ∅))) → ( 0s <s 1s ↔ (∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ âˆĻ ∃ð‘Ķ ∈ ∅ ð‘Ķ â‰Īs 1s )))
2211, 18, 19, 20, 21mp4an 690 . 2 ( 0s <s 1s ↔ (∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ âˆĻ ∃ð‘Ķ ∈ ∅ ð‘Ķ â‰Īs 1s ))
238, 22mpbir 230 1 0s <s 1s
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   âˆĻ wo 844   = wceq 1533   ∈ wcel 2098  âˆƒwrex 3064   ⊆ wss 3943  âˆ…c0 4317  ð’Ŧ cpw 4597  {csn 4623   class class class wbr 5141  (class class class)co 7404   No csur 27524   <s cslt 27525   â‰Īs csle 27628   <<s csslt 27664   |s cscut 27666   0s c0s 27706   1s c1s 27707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1o 8464  df-2o 8465  df-no 27527  df-slt 27528  df-bday 27529  df-sle 27629  df-sslt 27665  df-scut 27667  df-0s 27708  df-1s 27709
This theorem is referenced by:  left1s  27772  right1s  27773  divs1  28054  precsexlem9  28064  om2noseqlt  28123  n0scut  28154  n0sge0  28157  1nns  28166  nnsrecgt0d  28170  0reno  28176
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