Theorem cuteq0 28135

Description: Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)

Hypotheses

Ref	Expression
cuteq0.1	⊢ (𝜑 → 𝐴 <<s { 0s })
cuteq0.2	⊢ (𝜑 → { 0s } <<s 𝐵)

Assertion

Ref	Expression
cuteq0	⊢ (𝜑 → (𝐴 |s 𝐵) = 0s )

Proof of Theorem cuteq0

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

Step	Hyp	Ref	Expression
1		cuteq0.1	. 2 ⊢ (𝜑 → 𝐴 <<s { 0s })
2		cuteq0.2	. 2 ⊢ (𝜑 → { 0s } <<s 𝐵)
3		bday0 28131	. . 3 ⊢ ( bday ‘ 0s ) = ∅
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ( bday ‘ 0s ) = ∅)
5		0no 28129	. . . . . . 7 ⊢ 0s ∈ No
6		sneq 4593	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → {𝑦} = { 0s })
7	6	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
8	6	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ({𝑦} <<s 𝐵 ↔ { 0s } <<s 𝐵))
9	7, 8	anbi12d 644	. . . . . . . . 9 ⊢ (𝑦 = 0s → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵)))
10		fveqeq2 6882	. . . . . . . . 9 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
11	9, 10	anbi12d 644	. . . . . . . 8 ⊢ (𝑦 = 0s → (((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅) ↔ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)))
12	11	rspcev 3576	. . . . . . 7 ⊢ (( 0s ∈ No ∧ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)) → ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
13	5, 12	mpan 703	. . . . . 6 ⊢ (((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
14	1, 2, 4, 13	syl21anc 851	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
15		bdayfn 28068	. . . . . . 7 ⊢ bday Fn No
16		ssrab2 4027	. . . . . . 7 ⊢ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17		fvelimab 6945	. . . . . . 7 ⊢ (( bday Fn No ∧ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No ) → (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∅))
18	15, 16, 17	mp2an 705	. . . . . 6 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∅)
19		sneq 4593	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	breq2d 5114	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
21	19	breq1d 5112	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝐵 ↔ {𝑦} <<s 𝐵))
22	20, 21	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
23	22	rexrab 3653	. . . . . 6 ⊢ (∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∅ ↔ ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
24	18, 23	bitri 278	. . . . 5 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
25	14, 24	sylibr 237	. . . 4 ⊢ (𝜑 → ∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
26		int0el 4938	. . . 4 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) = ∅)
27	25, 26	syl 18	. . 3 ⊢ (𝜑 → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) = ∅)
28	3, 27	eqtr4id 2814	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
29	5	elexi 3472	. . . . . 6 ⊢ 0s ∈ V
30	29	snnz 4736	. . . . 5 ⊢ { 0s } ≠ ∅
31		sltstr 28107	. . . . 5 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ { 0s } ≠ ∅) → 𝐴 <<s 𝐵)
32	30, 31	mp3an3 1479	. . . 4 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) → 𝐴 <<s 𝐵)
33	1, 2, 32	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
34		eqcuts 28105	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 0s ∈ No ) → ((𝐴 |s 𝐵) = 0s ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))))
35	33, 5, 34	sylancl 598	. 2 ⊢ (𝜑 → ((𝐴 |s 𝐵) = 0s ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))))
36	1, 2, 28, 35	mpbir3and 1361	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 0s )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  {crab 3412   ⊆ wss 3898  ∅c0 4278  {csn 4583  ∩ cint 4906   class class class wbr 5102   “ cima 5650   Fn wfn 6522  ‘cfv 6527  (class class class)co 7408   No csur 27931   bday cbday 27933   <<s cslts 28077   |s ccuts 28079   0_s c0s 28125

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1o 8454  df-2o 8455  df-no 27934  df-lts 27935  df-bday 27936  df-slts 28078  df-cuts 28080  df-0s 28127

This theorem is used by:  cutneg  28136  negsid  28361