Theorem negsproplem5 27918

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	⊢ (𝜑 → 𝐴 ∈ No )
negsproplem4.2	⊢ (𝜑 → 𝐵 ∈ No )
negsproplem4.3	⊢ (𝜑 → 𝐴 <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.

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 27916	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
4	3	simp2d 1141	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5		negsfn 27910	. . 3 ⊢ -us Fn No
6		rightssno 27782	. . 3 ⊢ ( R ‘𝐴) ⊆ No
7		negsproplem5.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
8		bdayelon 27683	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
9		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
10		oldbday 27801	. . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 586	. . . . 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 5146	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <s 𝑏 ↔ 𝐴 <s 𝐵))
15		rightval 27765	. . . . 5 ⊢ ( R ‘𝐴) = {𝑏 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑏}
16	14, 15	elrab2 3683	. . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵))
17	12, 13, 16	sylanbrc 582	. . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴))
18		fnfvima 7239	. . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
19	5, 6, 17, 18	mp3an12i 1462	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
20		fvex 6904	. . . 4 ⊢ ( -us ‘𝐴) ∈ V
21	20	snid 4660	. . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
22	21	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
23	4, 19, 22	ssltsepcd 27701	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ∀wral 3056   ∪ cun 3942   ⊆ wss 3944  {csn 4624   class class class wbr 5142   “ cima 5675  Oncon0 6363   Fn wfn 6537  ‘cfv 6542   No csur 27547   <s cslt 27548   bday cbday 27549   <<s csslt 27687   O cold 27744   L cleft 27746   R cright 27747   -u_s 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:  negsproplem7  27920