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Theorem cuteq0 27122
Description: Condition for a surreal cut to equal zero. (Contributed by Scott Fenton, 3-Feb-2025.)
Hypotheses
Ref Expression
cuteq0.1 (𝜑𝐴 <<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.
StepHypRef Expression
1 cuteq0.1 . 2 (𝜑𝐴 <<s { 0s })
2 cuteq0.2 . 2 (𝜑 → { 0s } <<s 𝐵)
3 bday0s 27118 . . 3 ( bday ‘ 0s ) = ∅
43a1i 11 . . . . . 6 (𝜑 → ( bday ‘ 0s ) = ∅)
5 0sno 27116 . . . . . . 7 0s ∈ No
6 sneq 4594 . . . . . . . . . . 11 (𝑦 = 0s → {𝑦} = { 0s })
76breq2d 5115 . . . . . . . . . 10 (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
86breq1d 5113 . . . . . . . . . 10 (𝑦 = 0s → ({𝑦} <<s 𝐵 ↔ { 0s } <<s 𝐵))
97, 8anbi12d 631 . . . . . . . . 9 (𝑦 = 0s → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵)))
10 fveqeq2 6848 . . . . . . . . 9 (𝑦 = 0s → (( bday 𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
119, 10anbi12d 631 . . . . . . . 8 (𝑦 = 0s → (((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅) ↔ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)))
1211rspcev 3579 . . . . . . 7 (( 0s ∈ No ∧ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)) → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
135, 12mpan 688 . . . . . 6 (((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
141, 2, 4, 13syl21anc 836 . . . . 5 (𝜑 → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
15 bdayfn 27064 . . . . . . 7 bday Fn No
16 ssrab2 4035 . . . . . . 7 {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17 fvelimab 6911 . . . . . . 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) = ∅))
1815, 16, 17mp2an 690 . . . . . 6 (∅ ∈ ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ∅)
19 sneq 4594 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019breq2d 5115 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
2119breq1d 5113 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s 𝐵 ↔ {𝑦} <<s 𝐵))
2220, 21anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
2322rexrab 3652 . . . . . 6 (∃𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ∅ ↔ ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
2418, 23bitri 274 . . . . 5 (∅ ∈ ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
2514, 24sylibr 233 . . . 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)}) = ∅)
2725, 26syl 17 . . 3 (𝜑 ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) = ∅)
283, 27eqtr4id 2796 . 2 (𝜑 → ( bday ‘ 0s ) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
295elexi 3462 . . . . . 6 0s ∈ V
3029snnz 4735 . . . . 5 { 0s } ≠ ∅
31 sslttr 27097 . . . . 5 ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ { 0s } ≠ ∅) → 𝐴 <<s 𝐵)
3230, 31mp3an3 1450 . . . 4 ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) → 𝐴 <<s 𝐵)
331, 2, 32syl2anc 584 . . 3 (𝜑𝐴 <<s 𝐵)
34 eqscut 27095 . . 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)}))))
3533, 5, 34sylancl 586 . 2 (𝜑 → ((𝐴 |s 𝐵) = 0s ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ ( bday ‘ 0s ) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))))
361, 2, 28, 35mpbir3and 1342 1 (𝜑 → (𝐴 |s 𝐵) = 0s )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wrex 3071  {crab 3405  wss 3908  c0 4280  {csn 4584   cint 4905   class class class wbr 5103  cima 5634   Fn wfn 6488  cfv 6493  (class class class)co 7351   No csur 26939   bday cbday 26941   <<s csslt 27071   |s cscut 27073   0s c0s 27112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1o 8404  df-2o 8405  df-no 26942  df-slt 26943  df-bday 26944  df-sslt 27072  df-scut 27074  df-0s 27114
This theorem is referenced by:  negsid  34327
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