Theorem negsproplem5 28127

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 28125	. . 3 ⊢ (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
4	3	simp2d 1157	. 2 ⊢ (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5		negsfn 28118	. . 3 ⊢ -us Fn No
6		rightssno 27969	. . 3 ⊢ ( R ‘𝐴) ⊆ No
7		negsproplem5.4	. . . . 5 ⊢ (𝜑 → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
8		bdayon 27847	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
9		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
10		oldbday 27996	. . . . . 6 ⊢ ((( bday ‘𝐴) ∈ On ∧ 𝐵 ∈ No ) → (𝐵 ∈ ( O ‘( bday ‘𝐴)) ↔ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
11	8, 9, 10	sylancr 596	. . . . 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 27947	. . . 4 ⊢ (𝐵 ∈ ( R ‘𝐴) ↔ (𝐵 ∈ ( O ‘( bday ‘𝐴)) ∧ 𝐴 <s 𝐵))
15	12, 13, 14	sylanbrc 592	. . 3 ⊢ (𝜑 → 𝐵 ∈ ( R ‘𝐴))
16		fnfvima 7219	. . 3 ⊢ (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ∧ 𝐵 ∈ ( R ‘𝐴)) → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
17	5, 6, 15, 16	mp3an12i 1488	. 2 ⊢ (𝜑 → ( -us ‘𝐵) ∈ ( -us “ ( R ‘𝐴)))
18		fvex 6882	. . . 4 ⊢ ( -us ‘𝐴) ∈ V
19	18	snid 4623	. . 3 ⊢ ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
20	19	a1i 11	. 2 ⊢ (𝜑 → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
21	4, 17, 20	sltssepcd 27867	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2144  ∀wral 3078   ∪ cun 3904   ⊆ wss 3906  {csn 4584   class class class wbr 5102   “ cima 5652  Oncon0 6348   Fn wfn 6518  ‘cfv 6523   No csur 27706   <s clts 27707   bday cbday 27708   <<s cslts 27852   O cold 27918   L cleft 27920   R cright 27921   -u_s cnegs 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-1o 8439  df-2o 8440  df-no 27709  df-lts 27710  df-bday 27711  df-slts 27853  df-cuts 27855  df-0s 27902  df-made 27922  df-old 27923  df-left 27925  df-right 27926  df-norec 28033  df-negs 28116

This theorem is referenced by:  negsproplem7  28129