Theorem 0lt1s 28203

Description: Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)

Assertion

Ref	Expression
0lt1s	⊢ 0s <s 1s

Proof of Theorem 0lt1s

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0no 28200	. . . . 5 ⊢ 0s ∈ No
2		lesid 28129	. . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s )
3	1, 2	ax-mp 5	. . . 4 ⊢ 0s ≤s 0s
4	1	elexi 3472	. . . . 5 ⊢ 0s ∈ V
5		breq2 5106	. . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
6	4, 5	rexsn 4639	. . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s )
7	3, 6	mpbir 234	. . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥
8	7	orci 879	. 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )
9		0elpw 5316	. . . 4 ⊢ ∅ ∈ 𝒫 No
10		nulsgts 28167	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
11	9, 10	ax-mp 5	. . 3 ⊢ ∅ <<s ∅
12		snssi 4742	. . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No)
13	1, 12	ax-mp 5	. . . . 5 ⊢ { 0s } ⊆ No
14		snex 5396	. . . . . 6 ⊢ { 0s } ∈ V
15	14	elpw 4560	. . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No)
16	13, 15	mpbir 234	. . . 4 ⊢ { 0s } ∈ 𝒫 No
17		nulsgts 28167	. . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
18	16, 17	ax-mp 5	. . 3 ⊢ { 0s } <<s ∅
19		df-0s 28198	. . 3 ⊢ 0s = (∅ |s ∅)
20		df-1s 28199	. . 3 ⊢ 1s = ({ 0s } |s ∅)
21		ltsrec 28192	. . 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 706	. 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))
23	8, 22	mpbir 234	1 ⊢ 0s <s 1s

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102  (class class class)co 7420  Nocsur 28002   <s clts 28003   ≤s cles 28106   <<s cslts 28148   |s ccuts 28150   0_s c0s 28196   1_s c1s 28197

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199

This theorem is used by:  1ne0s  28211  left1s  28286  right1s  28287  ltsp1d  28406  precsexlem9  28606  n0sge0  28729  nnsrecgt0d  28742  twocut  28814  nohalf  28815  expsgt0  28828  pw2recs  28829  halfcut  28849  bdaypw2n0bndlem  28854  bdayfinbndlem1  28858  0reno  28887  1reno  28888