Theorem cuteq1 27976

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 27973	. . . . . 6 ⊢ ( bday ‘ 1s ) = 1o
4		df-1o 8453	. . . . . 6 ⊢ 1o = suc ∅
5	3, 4	eqtri 2792	. . . . 5 ⊢ ( bday ‘ 1s ) = suc ∅
6		sltssep 27926	. . . . . . . . . . . . . 14 ⊢ (𝐴 <<s { 0s } → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦)
7		dfral2 3122	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
8	7	ralbii 3117	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
9		ralnex 3097	. . . . . . . . . . . . . . 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 27968	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ No
14		ltsirr 27876	. . . . . . . . . . . . . . . 16 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
15	13, 14	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ¬ 0s <s 0s
16		breq1 5116	. . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . 15 ⊢ (( 0s ∈ 𝐴 ∧ ¬ 0s <s 0s ) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
19	12, 15, 18	sylancl 597	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
20	13	elexi 3485	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ V
21		breq2 5117	. . . . . . . . . . . . . . . . 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 4653	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s )
24	23	rexbii 3118	. . . . . . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ¬ 𝐴 <<s { 0s })
28		sneq 4604	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0s → {𝑥} = { 0s })
29	28	breq2d 5125	. . . . . . . . . . . 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 2979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} → 𝑥 ≠ 0s ))
33	32	adantrd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → 𝑥 ≠ 0s ))
34	33	impr 459	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → 𝑥 ≠ 0s )
35		bday0b 27972	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
36	35	ad2antrl 740	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
37	36	necon3bid 3008	. . . . . . 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 27911	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
40	39	onordi 6475	. . . . . . . 8 ⊢ Ord ( bday ‘𝑥)
41		ord0eln0 6418	. . . . . . . 8 ⊢ (Ord ( bday ‘𝑥) → (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅)
43		0elon 6417	. . . . . . . 8 ⊢ ∅ ∈ On
44	43, 39	onsucssi 7837	. . . . . . 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 3983	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥))
48	47	expr 461	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
49	48	ralrimiva 3163	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
50		1no 27969	. . . . . . 7 ⊢ 1s ∈ No
51	50	elexi 3485	. . . . . 6 ⊢ 1s ∈ V
52	51	snnz 4747	. . . . 5 ⊢ { 1s } ≠ ∅
53		sltstr 27946	. . . . 5 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ { 1s } ≠ ∅) → 𝐴 <<s 𝐵)
54	52, 53	mp3an3 1476	. . . 4 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵) → 𝐴 <<s 𝐵)
55	1, 2, 54	syl2anc 595	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
56		eqcuts2 27945	. . 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 597	. 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 1359	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 1s )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ⊆ wss 3913  ∅c0 4294  {csn 4594   class class class wbr 5113  Ord word 6360  suc csuc 6363  ‘cfv 6537  (class class class)co 7411  1_oc1o 8446   No csur 27770   <s clts 27771   bday cbday 27772   <<s cslts 27916   |s ccuts 27918   0_s c0s 27964   1_s c1s 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1o 8453  df-2o 8454  df-no 27773  df-lts 27774  df-bday 27775  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967

This theorem is referenced by:  precsexlem11  28376