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Theorem 0slt1s 27190
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 27187 . . . . 5 0s ∈ No
2 slerflex 27127 . . . . 5 ( 0s ∈ No → 0s â‰Īs 0s )
31, 2ax-mp 5 . . . 4 0s â‰Īs 0s
41elexi 3463 . . . . 5 0s ∈ V
5 breq2 5110 . . . . 5 (ð‘Ĩ = 0s → ( 0s â‰Īs ð‘Ĩ ↔ 0s â‰Īs 0s ))
64, 5rexsn 4644 . . . 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 864 . 2 (∃ð‘Ĩ ∈ { 0s } 0s â‰Īs ð‘Ĩ âˆĻ ∃ð‘Ķ ∈ ∅ ð‘Ķ â‰Īs 1s )
9 0elpw 5312 . . . 4 ∅ ∈ ð’Ŧ No
10 nulssgt 27159 . . . 4 (∅ ∈ ð’Ŧ No → ∅ <<s ∅)
119, 10ax-mp 5 . . 3 ∅ <<s ∅
12 snssi 4769 . . . . . 6 ( 0s ∈ No → { 0s } ⊆ No )
131, 12ax-mp 5 . . . . 5 { 0s } ⊆ No
14 snex 5389 . . . . . 6 { 0s } ∈ V
1514elpw 4565 . . . . 5 ({ 0s } ∈ ð’Ŧ No ↔ { 0s } ⊆ No )
1613, 15mpbir 230 . . . 4 { 0s } ∈ ð’Ŧ No
17 nulssgt 27159 . . . 4 ({ 0s } ∈ ð’Ŧ No → { 0s } <<s ∅)
1816, 17ax-mp 5 . . 3 { 0s } <<s ∅
19 df-0s 27185 . . 3 0s = (∅ |s ∅)
20 df-1s 27186 . . 3 1s = ({ 0s } |s ∅)
21 sltrec 27181 . . 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 692 . 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 846   = wceq 1542   ∈ wcel 2107  âˆƒwrex 3070   ⊆ wss 3911  âˆ…c0 4283  ð’Ŧ cpw 4561  {csn 4587   class class class wbr 5106  (class class class)co 7358   No csur 27004   <s cslt 27005   â‰Īs csle 27108   <<s csslt 27142   |s cscut 27144   0s c0s 27183   1s c1s 27184
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009  df-sle 27109  df-sslt 27143  df-scut 27145  df-0s 27185  df-1s 27186
This theorem is referenced by:  left1s  27246  right1s  27247
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