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Theorem negsproplem5 27957
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 27955 . . 3 (𝜑 → (( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))))
43simp2d 1140 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)})
5 negsfn 27949 . . 3 -us Fn No
6 rightssno 27821 . . 3 ( R ‘𝐴) ⊆ No
7 negsproplem5.4 . . . . 5 (𝜑 → ( bday 𝐵) ∈ ( bday 𝐴))
8 bdayelon 27722 . . . . . 6 ( bday 𝐴) ∈ On
9 negsproplem4.2 . . . . . 6 (𝜑𝐵 No )
10 oldbday 27840 . . . . . 6 ((( bday 𝐴) ∈ On ∧ 𝐵 No ) → (𝐵 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝐵) ∈ ( bday 𝐴)))
118, 9, 10sylancr 585 . . . . 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 5148 . . . . 5 (𝑏 = 𝐵 → (𝐴 <s 𝑏𝐴 <s 𝐵))
15 rightval 27804 . . . . 5 ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑏}
1614, 15elrab2 3679 . . . 4 (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday 𝐴)) ∧ 𝐴 <s 𝐵))
1712, 13, 16sylanbrc 581 . . 3 (𝜑𝐵 ∈ ( R ‘𝐴))
18 fnfvima 7239 . . 3 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No 𝐵 ∈ ( R ‘𝐴)) → ( -us𝐵) ∈ ( -us “ ( R ‘𝐴)))
195, 6, 17, 18mp3an12i 1461 . 2 (𝜑 → ( -us𝐵) ∈ ( -us “ ( R ‘𝐴)))
20 fvex 6903 . . . 4 ( -us𝐴) ∈ V
2120snid 4661 . . 3 ( -us𝐴) ∈ {( -us𝐴)}
2221a1i 11 . 2 (𝜑 → ( -us𝐴) ∈ {( -us𝐴)})
234, 19, 22ssltsepcd 27740 1 (𝜑 → ( -us𝐵) <s ( -us𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wral 3051  cun 3939  wss 3941  {csn 4625   class class class wbr 5144  cima 5676  Oncon0 6365   Fn wfn 6538  cfv 6543   No csur 27586   <s cslt 27587   bday cbday 27588   <<s csslt 27726   O cold 27783   L cleft 27785   R cright 27786   -us cnegs 27945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-no 27589  df-slt 27590  df-bday 27591  df-sslt 27727  df-scut 27729  df-0s 27770  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-negs 27947
This theorem is referenced by:  negsproplem7  27959
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