Theorem negsproplem5 28046

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

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 28044	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
4	3	simp2d 1150	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5		negsfn 28037	. . 3 ⊢ -us Fn No
6		rightssno 27888	. . 3 ⊢ ( R ‘𝐴) ⊆ No
7		negsproplem5.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
8		bdayon 27766	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
9		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
10		oldbday 27915	. . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 594	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
12	7, 11	mpbird 259	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ( O ‘( bday ‘𝐴)))
13		negsproplem4.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
14		elright 27866	. . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵))
15	12, 13, 14	sylanbrc 590	. . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴))
16		fnfvima 7181	. . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
17	5, 6, 15, 16	mp3an12i 1474	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
18		fvex 6844	. . . 4 ⊢ ( -us ‘𝐴) ∈ V
19	18	snid 4597	. . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
20	19	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
21	4, 17, 20	sltssepcd 27786	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∈ wcel 2121  ∀wral 3055   ∪ cun 3883   ⊆ wss 3885  {csn 4558   class class class wbr 5075   “ cima 5624  Oncon0 6314   Fn wfn 6484  ‘cfv 6489   No csur 27625   <s clts 27626   bday cbday 27627   <<s cslts 27771   O cold 27837   L cleft 27839   R cright 27840   -u_s cnegs 28033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  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 27628  df-lts 27629  df-bday 27630  df-slts 27772  df-cuts 27774  df-0s 27821  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-negs 28035

This theorem is referenced by:  negsproplem7  28048