Theorem negsproplem4 28063

Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (Contributed by Scott Fenton, 2-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)
negsproplem4.4	⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))

Assertion

Ref	Expression
negsproplem4	⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem4

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		negsproplem.1	. . . . 5 ⊢ (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		uncom 4158	. . . . . . . 8 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
3	2	eleq2i 2833	. . . . . . 7 ⊢ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)))
4	3	imbi1i 349	. . . . . 6 ⊢ (((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
5	4	2ralbii 3128	. . . . 5 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
6	1, 5	sylib 218	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐵) ∪ ( bday ‘𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
7		negsproplem4.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No )
8	6, 7	negsproplem3 28062	. . 3 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
9	8	simp3d 1145	. 2 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
10		fvex 6919	. . . 4 ⊢ ( -us ‘𝐵) ∈ V
11	10	snid 4662	. . 3 ⊢ ( -us ‘𝐵) ∈ {( -us ‘𝐵)}
12	11	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ {( -us ‘𝐵)})
13		negsfn 28055	. . 3 ⊢ -us Fn No
14		leftssno 27919	. . 3 ⊢ ( L ‘𝐵) ⊆ No
15		negsproplem4.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16		bdayelon 27821	. . . . . 6 ⊢ ( bday ‘𝐵) ∈ On
17		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
18		oldbday 27939	. . . . . 6 ⊢ ((( bday ‘𝐵) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
19	16, 17, 18	sylancr 587	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
20	15, 19	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵)))
21		negsproplem4.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
22		breq1 5146	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 <s 𝐵 ↔ 𝐴 <s 𝐵))
23		leftval 27902	. . . . 5 ⊢ ( L ‘𝐵) = {𝑎 ∈ ( O ‘( bday ‘𝐵)) ∣ 𝑎 <s 𝐵}
24	22, 23	elrab2 3695	. . . 4 ⊢ (𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵))
25	20, 21, 24	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵))
26		fnfvima 7253	. . 3 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝐴 ∈ ( L ‘𝐵)) → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
27	13, 14, 25, 26	mp3an12i 1467	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
28	9, 12, 27	ssltsepcd 27839	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

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   -u_s 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