Theorem 0slt1s 27741

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 27738	. . . . 5 ⊢ 0s ∈ No
2		slerflex 27675	. . . . 5 ⊢ ( 0s ∈ No → 0s ≤s 0s )
3	1, 2	ax-mp 5	. . . 4 ⊢ 0s ≤s 0s
4	1	elexi 3470	. . . . 5 ⊢ 0s ∈ V
5		breq2 5111	. . . . 5 ⊢ (𝑥 = 0s → ( 0s ≤s 𝑥 ↔ 0s ≤s 0s ))
6	4, 5	rexsn 4646	. . . 4 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ↔ 0s ≤s 0s )
7	3, 6	mpbir 231	. . 3 ⊢ ∃𝑥 ∈ { 0s } 0s ≤s 𝑥
8	7	orci 865	. 2 ⊢ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s )
9		0elpw 5311	. . . 4 ⊢ ∅ ∈ 𝒫 No
10		nulssgt 27710	. . . 4 ⊢ (∅ ∈ 𝒫 No → ∅ <<s ∅)
11	9, 10	ax-mp 5	. . 3 ⊢ ∅ <<s ∅
12		snssi 4772	. . . . . 6 ⊢ ( 0s ∈ No → { 0s } ⊆ No )
13	1, 12	ax-mp 5	. . . . 5 ⊢ { 0s } ⊆ No
14		snex 5391	. . . . . 6 ⊢ { 0s } ∈ V
15	14	elpw 4567	. . . . 5 ⊢ ({ 0s } ∈ 𝒫 No ↔ { 0s } ⊆ No )
16	13, 15	mpbir 231	. . . 4 ⊢ { 0s } ∈ 𝒫 No
17		nulssgt 27710	. . . 4 ⊢ ({ 0s } ∈ 𝒫 No → { 0s } <<s ∅)
18	16, 17	ax-mp 5	. . 3 ⊢ { 0s } <<s ∅
19		df-0s 27736	. . 3 ⊢ 0s = (∅ |s ∅)
20		df-1s 27737	. . 3 ⊢ 1s = ({ 0s } |s ∅)
21		sltrec 27732	. . 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 693	. 2 ⊢ ( 0s <s 1s ↔ (∃𝑥 ∈ { 0s } 0s ≤s 𝑥 ∨ ∃𝑦 ∈ ∅ 𝑦 ≤s 1s ))
23	8, 22	mpbir 231	1 ⊢ 0s <s 1s

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589   class class class wbr 5107  (class class class)co 7387   No csur 27551   <s cslt 27552   ≤s csle 27656   <<s csslt 27692   |s cscut 27694   0_s c0s 27734   1_s c1s 27735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1o 8434  df-2o 8435  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737

This theorem is referenced by:  1sne0s  27749  left1s  27806  right1s  27807  sltp1d  27922  precsexlem9  28117  n0sge0  28230  nnsrecgt0d  28243  twocut  28309  nohalf  28310  expsgt0  28322  pw2recs  28323  halfcut  28333  0reno  28348