Theorem cuteq0 27751

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		bday0s 27747	. . 3 ⊢ ( bday ‘ 0s ) = ∅
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ( bday ‘ 0s ) = ∅)
5		0sno 27745	. . . . . . 7 ⊢ 0s ∈ No
6		sneq 4602	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → {𝑦} = { 0s })
7	6	breq2d 5122	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
8	6	breq1d 5120	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ({𝑦} <<s 𝐵 ↔ { 0s } <<s 𝐵))
9	7, 8	anbi12d 632	. . . . . . . . 9 ⊢ (𝑦 = 0s → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵)))
10		fveqeq2 6870	. . . . . . . . 9 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
11	9, 10	anbi12d 632	. . . . . . . 8 ⊢ (𝑦 = 0s → (((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅) ↔ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)))
12	11	rspcev 3591	. . . . . . 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 690	. . . . . 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 837	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
15		bdayfn 27692	. . . . . . 7 ⊢ bday Fn No
16		ssrab2 4046	. . . . . . 7 ⊢ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17		fvelimab 6936	. . . . . . 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 692	. . . . . 6 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∅)
19		sneq 4602	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	breq2d 5122	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
21	19	breq1d 5120	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝐵 ↔ {𝑦} <<s 𝐵))
22	20, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
23	22	rexrab 3670	. . . . . 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 275	. . . . 5 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅))
25	14, 24	sylibr 234	. . . 4 ⊢ (𝜑 → ∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
26		int0el 4946	. . . 4 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) = ∅)
27	25, 26	syl 17	. . 3 ⊢ (𝜑 → ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) = ∅)
28	3, 27	eqtr4id 2784	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
29	5	elexi 3473	. . . . . 6 ⊢ 0s ∈ V
30	29	snnz 4743	. . . . 5 ⊢ { 0s } ≠ ∅
31		sslttr 27726	. . . . 5 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ { 0s } ≠ ∅) → 𝐴 <<s 𝐵)
32	30, 31	mp3an3 1452	. . . 4 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) → 𝐴 <<s 𝐵)
33	1, 2, 32	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
34		eqscut 27724	. . 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 586	. 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 1343	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 0s )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  {crab 3408   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∩ cint 4913   class class class wbr 5110   “ cima 5644   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   No csur 27558   bday cbday 27560   <<s csslt 27699   |s cscut 27701   0_s c0s 27741

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743

This theorem is referenced by:  cutneg  27752  negsid  27954  0reno  28355