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Theorem negsproplem5 34318
Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐵 is simpler than 𝐴. (Contributed by Scott Fenton, 3-Feb-2025.)
Hypotheses
Ref Expression
negsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
negsproplem4.1 (𝜑𝐴 No )
negsproplem4.2 (𝜑𝐵 No )
negsproplem4.3 (𝜑𝐴 <s 𝐵)
negsproplem5.4 (𝜑 → ( bday 𝐵) ∈ ( bday 𝐴))
Assertion
Ref Expression
negsproplem5 (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem5
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 negsproplem.1 . . . 4 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
2 negsproplem4.1 . . . 4 (𝜑𝐴 No )
31, 2negsproplem3 34316 . . 3 (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
43simp2d 1143 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5 negsfn 34310 . . 3 -us Fn No
6 rightssno 27161 . . 3 ( R ‘𝐴) ⊆ No
7 negsproplem5.4 . . . . 5 (𝜑 → ( bday 𝐵) ∈ ( bday 𝐴))
8 bdayelon 27067 . . . . . 6 ( bday 𝐴) ∈ On
9 negsproplem4.2 . . . . . 6 (𝜑𝐵 No )
10 oldbday 27178 . . . . . 6 ((( bday 𝐴) ∈ On ∧ 𝐵 No ) → (𝐵 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝐵) ∈ ( bday 𝐴)))
118, 9, 10sylancr 587 . . . . 5 (𝜑 → (𝐵 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝐵) ∈ ( bday 𝐴)))
127, 11mpbird 256 . . . 4 (𝜑𝐵 ∈ ( O ‘( bday 𝐴)))
13 negsproplem4.3 . . . 4 (𝜑𝐴 <s 𝐵)
14 breq2 5107 . . . . 5 (𝑏 = 𝐵 → (𝐴 <s 𝑏𝐴 <s 𝐵))
15 rightval 27145 . . . . 5 ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑏}
1614, 15elrab2 3646 . . . 4 (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday 𝐴)) ∧ 𝐴 <s 𝐵))
1712, 13, 16sylanbrc 583 . . 3 (𝜑𝐵 ∈ ( R ‘𝐴))
18 fnfvima 7179 . . 3 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
195, 6, 17, 18mp3an12i 1465 . 2 (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
20 fvex 6852 . . . 4 ( -us ‘𝐴) ∈ V
2120snid 4620 . . 3 ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
2221a1i 11 . 2 (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
234, 19, 22ssltsepcd 27084 1 (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wral 3062  cun 3906  wss 3908  {csn 4584   class class class wbr 5103  cima 5634  Oncon0 6315   Fn wfn 6488  cfv 6493   No csur 26939   <s cslt 26940   bday cbday 26941   <<s csslt 27071   O cold 27124   L cleft 27126   R cright 27127   -us cnegs 34306
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-pow 5318  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-se 5587  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-pred 6251  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-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-1o 8404  df-2o 8405  df-no 26942  df-slt 26943  df-bday 26944  df-sslt 27072  df-scut 27074  df-0s 27114  df-made 27128  df-old 27129  df-left 27131  df-right 27132  df-norec 34246  df-negs 34308
This theorem is referenced by:  negsproplem7  34320
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