Theorem 0lt1s 28228

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 28225	. . . . 5 ⊢ 0s ∈ No
2		lesid 28154	. . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s )
3	1, 2	ax-mp 5	. . . 4 ⊢ 0s ≤s 0s
4	1	elexi 3476	. . . . 5 ⊢ 0s ∈ V
5		breq2 5110	. . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
6	4, 5	rexsn 4643	. . . 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 5321	. . . 4 ⊢ ∅ ∈ 𝒫 No
10		nulsgts 28192	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
11	9, 10	ax-mp 5	. . 3 ⊢ ∅ <<s ∅
12		snssi 4746	. . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No)
13	1, 12	ax-mp 5	. . . . 5 ⊢ { 0s } ⊆ No
14		snex 5401	. . . . . 6 ⊢ { 0s } ∈ V
15	14	elpw 4564	. . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No)
16	13, 15	mpbir 234	. . . 4 ⊢ { 0s } ∈ 𝒫 No
17		nulsgts 28192	. . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
18	16, 17	ax-mp 5	. . 3 ⊢ { 0s } <<s ∅
19		df-0s 28223	. . 3 ⊢ 0s = (∅ |s ∅)
20		df-1s 28224	. . 3 ⊢ 1s = ({ 0s } |s ∅)
21		ltsrec 28217	. . 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 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5106  (class class class)co 7443  Nocsur 28027   <s clts 28028   ≤s cles 28131   <<s cslts 28173   |s ccuts 28175   0_s c0s 28221   1_s c1s 28222

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224

This theorem is used by:  1ne0s  28236  left1s  28311  right1s  28312  ltsp1d  28431  precsexlem9  28631  n0sge0  28754  nnsrecgt0d  28767  twocut  28839  nohalf  28840  expsgt0  28853  pw2recs  28854  halfcut  28874  bdaypw2n0bndlem  28879  bdayfinbndlem1  28883  0reno  28912  1reno  28913