Theorem cuteq1 27827

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 27824	. . . . . 6 ⊢ ( bday ‘ 1s ) = 1o
4		df-1o 8395	. . . . . 6 ⊢ 1o = suc ∅
5	3, 4	eqtri 2762	. . . . 5 ⊢ ( bday ‘ 1s ) = suc ∅
6		sltssep 27777	. . . . . . . . . . . . . 14 ⊢ (𝐴 <<s { 0s } → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦)
7		dfral2 3090	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
8	7	ralbii 3085	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
9		ralnex 3065	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
10	8, 9	bitri 276	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ { 0s }𝑥 <s 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
11	6, 10	sylib 219	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s { 0s } → ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
12		cuteq1.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0s ∈ 𝐴)
13		0no 27819	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ No
14		ltsirr 27728	. . . . . . . . . . . . . . . 16 ⊢ ( 0s ∈ No → ¬ 0s <s 0s )
15	13, 14	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ¬ 0s <s 0s
16		breq1 5075	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0s → (𝑥 <s 0s ↔ 0s <s 0s ))
17	16	notbid 319	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0s → (¬ 𝑥 <s 0s ↔ ¬ 0s <s 0s ))
18	17	rspcev 3560	. . . . . . . . . . . . . . 15 ⊢ (( 0s ∈ 𝐴 ∧ ¬ 0s <s 0s ) → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
19	12, 15, 18	sylancl 592	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
20	13	elexi 3453	. . . . . . . . . . . . . . . 16 ⊢ 0s ∈ V
21		breq2 5076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 0s → (𝑥 <s 𝑦 ↔ 𝑥 <s 0s ))
22	21	notbid 319	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 0s → (¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s ))
23	20, 22	rexsn 4614	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ¬ 𝑥 <s 0s )
24	23	rexbii 3086	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 <s 0s )
25	19, 24	sylibr 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ { 0s } ¬ 𝑥 <s 𝑦)
26	11, 25	nsyl3 138	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝐴 <<s { 0s })
27	26	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ¬ 𝐴 <<s { 0s })
28		sneq 4565	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0s → {𝑥} = { 0s })
29	28	breq2d 5084	. . . . . . . . . . . 12 ⊢ (𝑥 = 0s → (𝐴 <<s {𝑥} ↔ 𝐴 <<s { 0s }))
30	29	notbid 319	. . . . . . . . . . 11 ⊢ (𝑥 = 0s → (¬ 𝐴 <<s {𝑥} ↔ ¬ 𝐴 <<s { 0s }))
31	27, 30	syl5ibrcom 248	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → (𝑥 = 0s → ¬ 𝐴 <<s {𝑥}))
32	31	necon2ad 2949	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → (𝐴 <<s {𝑥} → 𝑥 ≠ 0s ))
33	32	adantrd 492	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → 𝑥 ≠ 0s ))
34	33	impr 455	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → 𝑥 ≠ 0s )
35		bday0b 27823	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
36	35	ad2antrl 734	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday ‘𝑥) = ∅ ↔ 𝑥 = 0s ))
37	36	necon3bid 2978	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → (( bday ‘𝑥) ≠ ∅ ↔ 𝑥 ≠ 0s ))
38	34, 37	mpbird 258	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘𝑥) ≠ ∅)
39		bdayon 27762	. . . . . . . . 9 ⊢ ( bday ‘𝑥) ∈ On
40	39	onordi 6423	. . . . . . . 8 ⊢ Ord ( bday ‘𝑥)
41		ord0eln0 6366	. . . . . . . 8 ⊢ (Ord ( bday ‘𝑥) → (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅))
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ (∅ ∈ ( bday ‘𝑥) ↔ ( bday ‘𝑥) ≠ ∅)
43		0elon 6365	. . . . . . . 8 ⊢ ∅ ∈ On
44	43, 39	onsucssi 7781	. . . . . . 7 ⊢ (∅ ∈ ( bday ‘𝑥) ↔ suc ∅ ⊆ ( bday ‘𝑥))
45	42, 44	bitr3i 278	. . . . . 6 ⊢ (( bday ‘𝑥) ≠ ∅ ↔ suc ∅ ⊆ ( bday ‘𝑥))
46	38, 45	sylib 219	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → suc ∅ ⊆ ( bday ‘𝑥))
47	5, 46	eqsstrid 3953	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ No ∧ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵))) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥))
48	47	expr 457	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ No ) → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
49	48	ralrimiva 3131	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) → ( bday ‘ 1s ) ⊆ ( bday ‘𝑥)))
50		1no 27820	. . . . . . 7 ⊢ 1s ∈ No
51	50	elexi 3453	. . . . . 6 ⊢ 1s ∈ V
52	51	snnz 4708	. . . . 5 ⊢ { 1s } ≠ ∅
53		sltstr 27797	. . . . 5 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵 ∧ { 1s } ≠ ∅) → 𝐴 <<s 𝐵)
54	52, 53	mp3an3 1458	. . . 4 ⊢ ((𝐴 <<s { 1s } ∧ { 1s } <<s 𝐵) → 𝐴 <<s 𝐵)
55	1, 2, 54	syl2anc 590	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
56		eqcuts2 27796	. . 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 592	. 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 1349	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 1s )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∅c0 4261  {csn 4555   class class class wbr 5072  Ord word 6309  suc csuc 6312  ‘cfv 6485  (class class class)co 7356  1_oc1o 8388   No csur 27621   <s clts 27622   bday cbday 27623   <<s cslts 27767   |s ccuts 27769   0_s c0s 27815   1_s c1s 27816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1o 8395  df-2o 8396  df-no 27624  df-lts 27625  df-bday 27626  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818

This theorem is referenced by:  precsexlem11  28227