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Theorem 0slt1s 33668
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 33665 . . . . 5 0s ∈ No
2 slerflex 33611 . . . . 5 ( 0s ∈ No → 0s ≤s 0s )
31, 2ax-mp 5 . . . 4 0s ≤s 0s
41elexi 3417 . . . . 5 0s ∈ V
5 breq2 5034 . . . . 5 (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
64, 5rexsn 4573 . . . 4 (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s )
73, 6mpbir 234 . . 3 𝑥 ∈ { 0s } 0s ≤s 𝑥
87orci 864 . 2 (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )
9 0elpw 5222 . . . 4 ∅ ∈ 𝒫 No
10 nulssgt 33637 . . . 4 (∅ ∈ 𝒫 No → ∅ <<s ∅)
119, 10ax-mp 5 . . 3 ∅ <<s ∅
12 snssi 4696 . . . . . 6 ( 0s ∈ No → { 0s } ⊆ No )
131, 12ax-mp 5 . . . . 5 { 0s } ⊆ No
14 snex 5298 . . . . . 6 { 0s } ∈ V
1514elpw 4492 . . . . 5 ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No )
1613, 15mpbir 234 . . . 4 { 0s } ∈ 𝒫 No
17 nulssgt 33637 . . . 4 ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
1816, 17ax-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 )))
2211, 18, 19, 20, 21mp4an 693 . 2 ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))
238, 22mpbir 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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