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| 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.) | 
| 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 | ⊢ (𝜑 → 𝐴 ∈ No ) | 
| negsproplem4.2 | ⊢ (𝜑 → 𝐵 ∈ No ) | 
| negsproplem4.3 | ⊢ (𝜑 → 𝐴 <s 𝐵) | 
| negsproplem5.4 | ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) | 
| Ref | Expression | 
|---|---|
| negsproplem5 | ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | 
| Step | Hyp | Ref | 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 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 3 | 1, 2 | negsproplem3 28062 | . . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴)))) | 
| 4 | 3 | simp2d 1144 | . 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)}) | 
| 5 | negsfn 28055 | . . 3 ⊢ -us Fn No | |
| 6 | rightssno 27920 | . . 3 ⊢ ( R ‘𝐴) ⊆ No | |
| 7 | negsproplem5.4 | . . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴)) | |
| 8 | bdayelon 27821 | . . . . . 6 ⊢ ( bday ‘𝐴) ∈ On | |
| 9 | negsproplem4.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No ) | |
| 10 | oldbday 27939 | . . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴))) | |
| 11 | 8, 9, 10 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴))) | 
| 12 | 7, 11 | mpbird 257 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ( O ‘( bday ‘𝐴))) | 
| 13 | negsproplem4.3 | . . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵) | |
| 14 | breq2 5147 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <s 𝑏 ↔ 𝐴 <s 𝐵)) | |
| 15 | rightval 27903 | . . . . 5 ⊢ ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑏} | |
| 16 | 14, 15 | elrab2 3695 | . . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵)) | 
| 17 | 12, 13, 16 | sylanbrc 583 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴)) | 
| 18 | fnfvima 7253 | . . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) | |
| 19 | 5, 6, 17, 18 | mp3an12i 1467 | . 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴))) | 
| 20 | fvex 6919 | . . . 4 ⊢ ( -us ‘𝐴) ∈ V | |
| 21 | 20 | snid 4662 | . . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)} | 
| 22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)}) | 
| 23 | 4, 19, 22 | ssltsepcd 27839 | 1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 ⊆ wss 3951 {csn 4626 class class class wbr 5143 “ cima 5688 Oncon0 6384 Fn wfn 6556 ‘cfv 6561 No csur 27684 <s cslt 27685 bday cbday 27686 <<s csslt 27825 O cold 27882 L cleft 27884 R cright 27885 -us cnegs 28051 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-0s 27869 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-negs 28053 | 
| This theorem is referenced by: negsproplem7 28066 | 
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