Theorem negsproplem4 27944

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

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 4124	. . . . . . . 8 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
3	2	eleq2i 2821	. . . . . . 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 3109	. . . . 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 27943	. . 3 ⊢ (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
9	8	simp3d 1144	. 2 ⊢ (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
10		fvex 6874	. . . 4 ⊢ ( -us ‘𝐵) ∈ V
11	10	snid 4629	. . 3 ⊢ ( -us ‘𝐵) ∈ {( -us ‘𝐵)}
12	11	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ {( -us ‘𝐵)})
13		negsfn 27936	. . 3 ⊢ -us Fn No
14		leftssno 27799	. . 3 ⊢ ( L ‘𝐵) ⊆ No
15		negsproplem4.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16		bdayelon 27695	. . . . . 6 ⊢ ( bday ‘𝐵) ∈ On
17		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
18		oldbday 27819	. . . . . 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		elleft 27780	. . . 4 ⊢ (𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵))
23	20, 21, 22	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵))
24		fnfvima 7210	. . 3 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝐴 ∈ ( L ‘𝐵)) → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
25	13, 14, 23, 24	mp3an12i 1467	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
26	9, 12, 25	ssltsepcd 27713	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∀wral 3045   ∪ cun 3915   ⊆ wss 3917  {csn 4592   class class class wbr 5110   “ cima 5644  Oncon0 6335   Fn wfn 6509  ‘cfv 6514   No csur 27558   <s cslt 27559   bday cbday 27560   <<s csslt 27699   O cold 27758   L cleft 27760   R cright 27761   -u_s cnegs 27932

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-negs 27934

This theorem is referenced by:  negsproplem7  27947