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Theorem cuteq0 27892
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 27888 . . 3 ( bday ‘ 0s ) = ∅
43a1i 11 . . . . . 6 (𝜑 → ( bday ‘ 0s ) = ∅)
5 0sno 27886 . . . . . . 7 0s No
6 sneq 4641 . . . . . . . . . . 11 (𝑦 = 0s → {𝑦} = { 0s })
76breq2d 5160 . . . . . . . . . 10 (𝑦 = 0s → (𝐴 <<s {𝑦} ↔ 𝐴 <<s { 0s }))
86breq1d 5158 . . . . . . . . . 10 (𝑦 = 0s → ({𝑦} <<s 𝐵 ↔ { 0s } <<s 𝐵))
97, 8anbi12d 632 . . . . . . . . 9 (𝑦 = 0s → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵)))
10 fveqeq2 6916 . . . . . . . . 9 (𝑦 = 0s → (( bday 𝑦) = ∅ ↔ ( bday ‘ 0s ) = ∅))
119, 10anbi12d 632 . . . . . . . 8 (𝑦 = 0s → (((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅) ↔ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)))
1211rspcev 3622 . . . . . . 7 (( 0s No ∧ ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅)) → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
135, 12mpan 690 . . . . . 6 (((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) ∧ ( bday ‘ 0s ) = ∅) → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
141, 2, 4, 13syl21anc 838 . . . . 5 (𝜑 → ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
15 bdayfn 27833 . . . . . . 7 bday Fn No
16 ssrab2 4090 . . . . . . 7 {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ⊆ No
17 fvelimab 6981 . . . . . . 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 692 . . . . . 6 (∅ ∈ ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ∅)
19 sneq 4641 . . . . . . . . 9 (𝑥 = 𝑦 → {𝑥} = {𝑦})
2019breq2d 5160 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴 <<s {𝑥} ↔ 𝐴 <<s {𝑦}))
2119breq1d 5158 . . . . . . . 8 (𝑥 = 𝑦 → ({𝑥} <<s 𝐵 ↔ {𝑦} <<s 𝐵))
2220, 21anbi12d 632 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
2322rexrab 3705 . . . . . 6 (∃𝑦 ∈ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)} ( bday 𝑦) = ∅ ↔ ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
2418, 23bitri 275 . . . . 5 (∅ ∈ ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}) ↔ ∃𝑦 No ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ∧ ( bday 𝑦) = ∅))
2514, 24sylibr 234 . . . 4 (𝜑 → ∅ ∈ ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
26 int0el 4984 . . . 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 2794 . 2 (𝜑 → ( bday ‘ 0s ) = ( bday “ {𝑥 No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)}))
295elexi 3501 . . . . . 6 0s ∈ V
3029snnz 4781 . . . . 5 { 0s } ≠ ∅
31 sslttr 27867 . . . . 5 ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵 ∧ { 0s } ≠ ∅) → 𝐴 <<s 𝐵)
3230, 31mp3an3 1449 . . . 4 ((𝐴 <<s { 0s } ∧ { 0s } <<s 𝐵) → 𝐴 <<s 𝐵)
331, 2, 32syl2anc 584 . . 3 (𝜑𝐴 <<s 𝐵)
34 eqscut 27865 . . 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 1341 1 (𝜑 → (𝐴 |s 𝐵) = 0s )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  wss 3963  c0 4339  {csn 4631   cint 4951   class class class wbr 5148  cima 5692   Fn wfn 6558  cfv 6563  (class class class)co 7431   No csur 27699   bday cbday 27701   <<s csslt 27840   |s cscut 27842   0s c0s 27882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884
This theorem is referenced by:  negsid  28088  n0scut  28353  0reno  28444
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