Theorem 0slt1s 27780

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.

Step	Hyp	Ref	Expression
1		0sno 27777	. . . . 5 ⊢ 0s ∈ No
2		slerflex 27714	. . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s )
3	1, 2	ax-mp 5	. . . 4 ⊢ 0s ≤s 0s
4	1	elexi 3491	. . . . 5 ⊢ 0s ∈ V
5		breq2 5154	. . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
6	4, 5	rexsn 4689	. . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s )
7	3, 6	mpbir 230	. . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥
8	7	orci 863	. 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )
9		0elpw 5358	. . . 4 ⊢ ∅ ∈ 𝒫 No
10		nulssgt 27749	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
11	9, 10	ax-mp 5	. . 3 ⊢ ∅ <<s ∅
12		snssi 4814	. . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No )
13	1, 12	ax-mp 5	. . . . 5 ⊢ { 0s } ⊆ No
14		snex 5435	. . . . . 6 ⊢ { 0s } ∈ V
15	14	elpw 4608	. . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No )
16	13, 15	mpbir 230	. . . 4 ⊢ { 0s } ∈ 𝒫 No
17		nulssgt 27749	. . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
18	16, 17	ax-mp 5	. . 3 ⊢ { 0s } <<s ∅
19		df-0s 27775	. . 3 ⊢ 0s = (∅ |s ∅)
20		df-1s 27776	. . 3 ⊢ 1s = ({ 0s } |s ∅)
21		sltrec 27771	. . 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 691	. 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))
23	8, 22	mpbir 230	1 ⊢ 0s <s 1s

Syntax hints:   ↔ wb 205   ∨ wo 845   = wceq 1533   ∈ wcel 2098  ∃wrex 3066   ⊆ wss 3947  ∅c0 4324  𝒫 cpw 4604  {csn 4630   class class class wbr 5150  (class class class)co 7424   No csur 27591   <s cslt 27592   ≤s csle 27695   <<s csslt 27731   |s cscut 27733   0_s c0s 27773   1_s c1s 27774

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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1o 8491  df-2o 8492  df-no 27594  df-slt 27595  df-bday 27596  df-sle 27696  df-sslt 27732  df-scut 27734  df-0s 27775  df-1s 27776

This theorem is referenced by:  left1s  27839  right1s  27840  divs1  28121  precsexlem9  28131  om2noseqlt  28190  n0scut  28221  n0sge0  28224  1nns  28233  nnsrecgt0d  28237  0reno  28243