Theorem cuteq0 27776

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 27772	. . 3 ⊢ ( bday ‘ 0s ) = ∅
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ( bday ‘ 0s ) = ∅)
5		0sno 27770	. . . . . . 7 ⊢ 0s ∈ No
6		sneq 4583	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → {𝑦} = { 0s })
7	6	breq2d 5101	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
8	6	breq1d 5099	. . . . . . . . . 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 6831	. . . . . . . . 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 3572	. . . . . . 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 27712	. . . . . . 7 ⊢ bday Fn No
16		ssrab2 4027	. . . . . . 7 ⊢ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17		fvelimab 6894	. . . . . . 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 4583	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	breq2d 5101	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
21	19	breq1d 5099	. . . . . . . 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 3650	. . . . . 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 4927	. . . 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 2785	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
29	5	elexi 3459	. . . . . 6 ⊢ 0s ∈ V
30	29	snnz 4726	. . . . 5 ⊢ { 0s } ≠ ∅
31		sslttr 27748	. . . . 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 27746	. . 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 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056  {crab 3395   ⊆ wss 3897  ∅c0 4280  {csn 4573  ∩ cint 4895   class class class wbr 5089   “ cima 5617   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   No csur 27578   bday cbday 27580   <<s csslt 27720   |s cscut 27722   0_s c0s 27766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sslt 27721  df-scut 27723  df-0s 27768

This theorem is referenced by:  cutneg  27777  negsid  27983  0reno  28399