Theorem 0slt1s 27319

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 27316	. . . . 5 ⊢ 0s ∈ No
2		slerflex 27255	. . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s )
3	1, 2	ax-mp 5	. . . 4 ⊢ 0s ≤s 0s
4	1	elexi 3493	. . . . 5 ⊢ 0s ∈ V
5		breq2 5151	. . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
6	4, 5	rexsn 4685	. . . 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 5353	. . . 4 ⊢ ∅ ∈ 𝒫 No
10		nulssgt 27288	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
11	9, 10	ax-mp 5	. . 3 ⊢ ∅ <<s ∅
12		snssi 4810	. . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No )
13	1, 12	ax-mp 5	. . . . 5 ⊢ { 0s } ⊆ No
14		snex 5430	. . . . . 6 ⊢ { 0s } ∈ V
15	14	elpw 4605	. . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No )
16	13, 15	mpbir 230	. . . 4 ⊢ { 0s } ∈ 𝒫 No
17		nulssgt 27288	. . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
18	16, 17	ax-mp 5	. . 3 ⊢ { 0s } <<s ∅
19		df-0s 27314	. . 3 ⊢ 0s = (∅ |s ∅)
20		df-1s 27315	. . 3 ⊢ 1s = ({ 0s } |s ∅)
21		sltrec 27310	. . 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 1541   ∈ wcel 2106  ∃wrex 3070   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  (class class class)co 7405   No csur 27132   <s cslt 27133   ≤s csle 27236   <<s csslt 27271   |s cscut 27273   0_s c0s 27312   1_s c1s 27313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1o 8462  df-2o 8463  df-no 27135  df-slt 27136  df-bday 27137  df-sle 27237  df-sslt 27272  df-scut 27274  df-0s 27314  df-1s 27315

This theorem is referenced by:  left1s  27378  right1s  27379  divs1  27640  precsexlem9  27650