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Theorem negsproplem3 27916
Description: Lemma for surreal negation. Give the cut properties of surreal negation. (Contributed by Scott Fenton, 2-Feb-2025.)
Hypotheses
Ref Expression
negsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
negsproplem2.1 (𝜑𝐴 No )
Assertion
Ref Expression
negsproplem3 (𝜑 → (( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem3
StepHypRef Expression
1 negsproplem.1 . . . 4 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
2 negsproplem2.1 . . . 4 (𝜑𝐴 No )
31, 2negsproplem2 27915 . . 3 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
4 scutcut 27708 . . 3 (( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)) → ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
53, 4syl 17 . 2 (𝜑 → ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
6 negsval 27912 . . . . 5 (𝐴 No → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
72, 6syl 17 . . . 4 (𝜑 → ( -us𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
87eleq1d 2813 . . 3 (𝜑 → (( -us𝐴) ∈ No ↔ (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ))
97sneqd 4636 . . . 4 (𝜑 → {( -us𝐴)} = {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))})
109breq2d 5154 . . 3 (𝜑 → (( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ↔ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))}))
119breq1d 5152 . . 3 (𝜑 → ({( -us𝐴)} <<s ( -us “ ( L ‘𝐴)) ↔ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴))))
128, 10, 113anbi123d 1433 . 2 (𝜑 → ((( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))) ↔ ((( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} ∧ {(( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))} <<s ( -us “ ( L ‘𝐴)))))
135, 12mpbird 257 1 (𝜑 → (( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  cun 3942  {csn 4624   class class class wbr 5142  cima 5675  cfv 6542  (class class class)co 7414   No csur 27547   <s cslt 27548   bday cbday 27549   <<s csslt 27687   |s cscut 27689   L cleft 27746   R cright 27747   -us cnegs 27906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-no 27550  df-slt 27551  df-bday 27552  df-sslt 27688  df-scut 27690  df-0s 27731  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec 27829  df-negs 27908
This theorem is referenced by:  negsproplem4  27917  negsproplem5  27918  negsproplem6  27919  negsprop  27921  negscut  27925
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