Theorem negsproplem4 28031

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 4111	. . . . . . . 8 ⊢ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) = (( bday ‘𝐵) ∪ ( bday ‘𝐴))
3	2	eleq2i 2829	. . . . . . 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 3112	. . . . 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 28030	. . 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 6848	. . . 4 ⊢ ( -us ‘𝐵) ∈ V
11	10	snid 4620	. . 3 ⊢ ( -us ‘𝐵) ∈ {( -us ‘𝐵)}
12	11	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ {( -us ‘𝐵)})
13		negsfn 28023	. . 3 ⊢ -us Fn No
14		leftssno 27873	. . 3 ⊢ ( L ‘𝐵) ⊆ No
15		negsproplem4.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16		bdayon 27752	. . . . . 6 ⊢ ( bday ‘𝐵) ∈ On
17		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
18		oldbday 27901	. . . . . 6 ⊢ ((( bday ‘𝐵) ∈ On ∧ 𝐴 ∈ No ) → (𝐴 ∈ ( O ‘( bday ‘𝐵)) ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
19	16, 17, 18	sylancr 588	. . . . 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 27851	. . . 4 ⊢ (𝐴 ∈ ( L ‘𝐵) ↔ (𝐴 ∈ ( O ‘( bday ‘𝐵)) ∧ 𝐴 <s 𝐵))
23	20, 21, 22	sylanbrc 584	. . 3 ⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵))
24		fnfvima 7181	. . 3 ⊢ (( -us Fn No ∧ ( L ‘𝐵) ⊆ No ∧ 𝐴 ∈ ( L ‘𝐵)) → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
25	13, 14, 23, 24	mp3an12i 1468	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ ( -us “ ( L ‘𝐵)))
26	9, 12, 25	sltssepcd 27772	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   ∪ cun 3900   ⊆ wss 3902  {csn 4581   class class class wbr 5099   “ cima 5628  Oncon0 6318   Fn wfn 6488  ‘cfv 6493   No csur 27611   <s clts 27612   bday cbday 27613   <<s cslts 27757   O cold 27823   L cleft 27825   R cright 27826   -u_s cnegs 28019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27614  df-lts 27615  df-bday 27616  df-slts 27758  df-cuts 27760  df-0s 27807  df-made 27827  df-old 27828  df-left 27830  df-right 27831  df-norec 27938  df-negs 28021

This theorem is referenced by:  negsproplem7  28034