Theorem cuteq0 27895

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 27891	. . 3 ⊢ ( bday ‘ 0s ) = ∅
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ( bday ‘ 0s ) = ∅)
5		0sno 27889	. . . . . . 7 ⊢ 0s ∈ No
6		sneq 4658	. . . . . . . . . . 11 ⊢ (𝑦 = 0s → {𝑦} = { 0s })
7	6	breq2d 5178	. . . . . . . . . 10 ⊢ (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
8	6	breq1d 5176	. . . . . . . . . 10 ⊢ (𝑦 = 0s → ({𝑦} <<s 𝐵 ↔ { 0s } <<s 𝐵))
9	7, 8	anbi12d 631	. . . . . . . . 9 ⊢ (𝑦 = 0s → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵)))
10		fveqeq2 6929	. . . . . . . . 9 ⊢ (𝑦 = 0s → (( bday ‘𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
11	9, 10	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = 0s → (((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday ‘𝑦) = ∅) ↔ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)))
12	11	rspcev 3635	. . . . . . 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 689	. . . . . 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 27836	. . . . . . 7 ⊢ bday Fn No
16		ssrab2 4103	. . . . . . 7 ⊢ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17		fvelimab 6994	. . . . . . 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 691	. . . . . 6 ⊢ (∅ ∈ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday ‘𝑦) = ∅)
19		sneq 4658	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
20	19	breq2d 5178	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
21	19	breq1d 5176	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ({𝑥} <<s 𝐵 ↔ {𝑦} <<s 𝐵))
22	20, 21	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
23	22	rexrab 3718	. . . . . 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 5003	. . . 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 2799	. 2 ⊢ (𝜑 → ( bday ‘ 0s ) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
29	5	elexi 3511	. . . . . 6 ⊢ 0s ∈ V
30	29	snnz 4801	. . . . 5 ⊢ { 0s } ≠ ∅
31		sslttr 27870	. . . . 5 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ { 0s } ≠ ∅) → 𝐴 <<s 𝐵)
32	30, 31	mp3an3 1450	. . . 4 ⊢ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) → 𝐴 <<s 𝐵)
33	1, 2, 32	syl2anc 583	. . 3 ⊢ (𝜑 → 𝐴 <<s 𝐵)
34		eqscut 27868	. . 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 585	. 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 1342	1 ⊢ (𝜑 → (𝐴 |s 𝐵) = 0s )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∩ cint 4970   class class class wbr 5166   “ cima 5703   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448   No csur 27702   bday cbday 27704   <<s csslt 27843   |s cscut 27845   0_s c0s 27885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887

This theorem is referenced by:  negsid  28091  n0scut  28356  0reno  28447