Theorem cuteq1 28063

Description: Condition for a surreal cut to equal one. (Contributed by Scott Fenton, 12-Mar-2025.)

Hypotheses

Ref	Expression
cuteq1.1	⊢ (𝜑 → 0s ∈ 𝐴)
cuteq1.2	⊢ (𝜑 → 𝐴 <<s { 1s })
cuteq1.3	⊢ (𝜑 → { 1s } <<s 𝐵)

Assertion

Ref	Expression
cuteq1	⊢ (𝜑 → (𝐴 |s 𝐵) = 1s )

Proof of Theorem cuteq1

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

Step	Hyp	Ref	Expression
1		cuteq1.2	. 2 ⊢ (𝜑 → 𝐴 <<s { 1s })
2		cuteq1.3	. 2 ⊢ (𝜑 → { 1s } <<s 𝐵)
3		bday1 28060	. . . . . 6 ⊢ ( bday ‘ 1s ) = 1o
4		df-1o 8459	. . . . . 6 ⊢ 1o = suc ∅
5	3, 4	eqtri 2788	. . . . 5 ⊢ ( bday ‘ 1s ) = suc ∅
6		sltssep 28013	. . . . . . . . . . . . . 14 ⊢ (𝐴 <<s { 0s } → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦)
7		dfral2 3118	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
8	7	ralbii 3113	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
9		ralnex 3093	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
10	8, 9	bitri 278	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
11	6, 10	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s { 0s } → ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
12		cuteq1.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0s ∈ 𝐴)
13		0no 28055	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ No
14		ltsirr 27963	. . . . . . . . . . . . . . . 16 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
15	13, 14	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ¬ 0s <s 0s
16		breq1 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0s → (𝑥 <s 0s ↔ 0s <s 0s ))
17	16	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0s → (¬ 𝑥 <s 0s ↔ ¬ 0s <s 0s ))
18	17	rspcev 3583	. . . . . . . . . . . . . . 15 ⊢ (( 0s ∈ 𝐴 ∧ ¬ 0s <s 0s ) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
19	12, 15, 18	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
20	13	elexi 3479	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ V
21		breq2 5115	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 0s → (𝑥 <s 𝑦 ↔ 𝑥 <s 0s ))
22	21	notbid 321	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0s → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s ))
23	20, 22	rexsn 4650	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s )
24	23	rexbii 3114	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
25	19, 24	sylibr 237	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
26	11, 25	nsyl3 139	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝐴 <<s { 0s })
27	26	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ¬ 𝐴 <<s { 0s })
28		sneq 4601	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0s → {𝑥} = { 0s })
29	28	breq2d 5123	. . . . . . . . . . . 12 ⊢ (𝑥 = 0s → (𝐴 <<s {𝑥} ↔ 𝐴 <<s { 0s }))
30	29	notbid 321	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (¬ 𝐴 <<s {𝑥} ↔ ¬ 𝐴 <<s { 0s }))
31	27, 30	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → (𝑥 = 0s → ¬ 𝐴 <<s {𝑥}))
32	31	necon2ad 2975	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} → 𝑥 ≠ 0s ))
33	32	adantrd 497	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → 𝑥 ≠ 0s ))
34	33	impr 460	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → 𝑥 ≠ 0s )
35		bday0b 28059	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
36	35	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
37	36	necon3bid 3004	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday ‘𝑥) ≠ ∅ ↔ 𝑥 ≠ 0s ))
38	34, 37	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘𝑥) ≠ ∅)
39		bdayon 27998	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
40	39	onordi 6478	. . . . . . . 8 ⊢ Ord ( bday ‘𝑥)
41		ord0eln0 6421	. . . . . . . 8 ⊢ (Ord ( bday ‘𝑥) → (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅)
43		0elon 6420	. . . . . . . 8 ⊢ ∅ ∈ On
44	43, 39	onsucssi 7843	. . . . . . 7 ⊢ (∅ ∈ ( bday ‘𝑥) ↔ suc ∅ ⊆ ( bday ‘𝑥))
45	42, 44	bitr3i 280	. . . . . 6 ⊢ (( bday ‘𝑥) ≠ ∅ ↔ suc ∅ ⊆ ( bday ‘𝑥))
46	38, 45	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → suc ∅ ⊆ ( bday ‘𝑥))
47	5, 46	eqsstrid 3976	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥))
48	47	expr 462	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
49	48	ralrimiva 3159	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
50		1no 28056	. . . . . . 7 ⊢ 1s ∈ No
51	50	elexi 3479	. . . . . 6 ⊢ 1s ∈ V
52	51	snnz 4744	. . . . 5 ⊢ { 1s } ≠ ∅
53		sltstr 28033	. . . . 5 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ { 1s } ≠ ∅) → 𝐴 <<s 𝐵)
54	52, 53	mp3an3 1479	. . . 4 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵) → 𝐴 <<s 𝐵)
55	1, 2, 54	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
56		eqcuts2 28032	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 1s ∈ No ) → ((𝐴 |s 𝐵) = 1s ↔ (𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ ∀𝑥 ∈ No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))))
57	55, 50, 56	sylancl 598	. 2 ⊢ (𝜑 → ((𝐴 |s 𝐵) = 1s ↔ (𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ ∀𝑥 ∈ No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))))
58	1, 2, 49, 57	mpbir3and 1361	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 1s )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∅c0 4286  {csn 4591   class class class wbr 5111  Ord word 6363  suc csuc 6366  ‘cfv 6540  (class class class)co 7419  1_oc1o 8452   No csur 27857   <s clts 27858   bday cbday 27859   <<s cslts 28003   |s ccuts 28005   0_s c0s 28051   1_s c1s 28052

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1o 8459  df-2o 8460  df-no 27860  df-lts 27861  df-bday 27862  df-slts 28004  df-cuts 28006  df-0s 28053  df-1s 28054

This theorem is used by:  precsexlem11  28463