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Theorem negsproplem5 28079
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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
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 → ( -us𝐵) <s ( -us𝐴))
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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
2 negsproplem4.1 . . . 4 (𝜑𝐴 No )
31, 2negsproplem3 28077 . . 3 (𝜑 → (( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))))
43simp2d 1142 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)})
5 negsfn 28070 . . 3 -us Fn No
6 rightssno 27935 . . 3 ( R ‘𝐴) ⊆ No
7 negsproplem5.4 . . . . 5 (𝜑 → ( bday 𝐵) ∈ ( bday 𝐴))
8 bdayelon 27836 . . . . . 6 ( bday 𝐴) ∈ On
9 negsproplem4.2 . . . . . 6 (𝜑𝐵 No )
10 oldbday 27954 . . . . . 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 257 . . . 4 (𝜑𝐵 ∈ ( O ‘( bday 𝐴)))
13 negsproplem4.3 . . . 4 (𝜑𝐴 <s 𝐵)
14 breq2 5152 . . . . 5 (𝑏 = 𝐵 → (𝐴 <s 𝑏𝐴 <s 𝐵))
15 rightval 27918 . . . . 5 ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑏}
1614, 15elrab2 3698 . . . 4 (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday 𝐴)) ∧ 𝐴 <s 𝐵))
1712, 13, 16sylanbrc 583 . . 3 (𝜑𝐵 ∈ ( R ‘𝐴))
18 fnfvima 7253 . . 3 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No 𝐵 ∈ ( R ‘𝐴)) → ( -us𝐵) ∈ ( -us “ ( R ‘𝐴)))
195, 6, 17, 18mp3an12i 1464 . 2 (𝜑 → ( -us𝐵) ∈ ( -us “ ( R ‘𝐴)))
20 fvex 6920 . . . 4 ( -us𝐴) ∈ V
2120snid 4667 . . 3 ( -us𝐴) ∈ {( -us𝐴)}
2221a1i 11 . 2 (𝜑 → ( -us𝐴) ∈ {( -us𝐴)})
234, 19, 22ssltsepcd 27854 1 (𝜑 → ( -us𝐵) <s ( -us𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wral 3059  cun 3961  wss 3963  {csn 4631   class class class wbr 5148  cima 5692  Oncon0 6386   Fn wfn 6558  cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   O cold 27897   L cleft 27899   R cright 27900   -us cnegs 28066
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-iun 4998  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-se 5642  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-pred 6323  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-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-negs 28068
This theorem is referenced by:  negsproplem7  28081
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