Theorem negsproplem4 28444

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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘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 → ( -s‘𝐵) <s ( -s‘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)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
2		uncom 4160	. . . . . . . 8 ⊢ ((bday‘𝐴) ∪ (bday‘𝐵)) = ((bday‘𝐵) ∪ (bday‘𝐴))
3	2	eleq2i 2856	. . . . . . 7 ⊢ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) ↔ ((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)))
4	3	imbi1i 352	. . . . . 6 ⊢ ((((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
5	4	2ralbii 3141	. . . . 5 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
6	1, 5	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
7		negsproplem4.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No)
8	6, 7	negsproplem3 28443	. . 3 ⊢ (𝜑 → (( -s‘𝐵) ∈ No ∧ ( -s “ (R‘𝐵)) <<s {( -s‘𝐵)} ∧ {( -s‘𝐵)} <<s ( -s “ (L‘𝐵))))
9	8	simp3d 1162	. 2 ⊢ (𝜑 → {( -s‘𝐵)} <<s ( -s “ (L‘𝐵)))
10		fvex 6918	. . . 4 ⊢ ( -s‘𝐵) ∈ V
11	10	snid 4623	. . 3 ⊢ ( -s‘𝐵) ∈ {( -s‘𝐵)}
12	11	a1i 11	. 2 ⊢ (𝜑 → ( -s‘𝐵) ∈ {( -s‘𝐵)})
13		negsfn 28436	. . 3 ⊢ -s Fn No
14		leftssno 28286	. . 3 ⊢ (L‘𝐵) ⊆ No
15		negsproplem4.4	. . . . 5 ⊢ (𝜑 → (bday‘𝐴) ∈ (bday‘𝐵))
16		bdayon 28165	. . . . . 6 ⊢ (bday‘𝐵) ∈ On
17		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No)
18		oldbday 28314	. . . . . 6 ⊢ (((bday‘𝐵) ∈ On ∧ 𝐴 ∈ No) → (𝐴 ∈ (O‘(bday‘𝐵)) ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
19	16, 17, 18	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (O‘(bday‘𝐵)) ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
20	15, 19	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (O‘(bday‘𝐵)))
21		negsproplem4.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
22		elleft 28264	. . . 4 ⊢ (𝐴 ∈ (L‘𝐵) ↔ (𝐴 ∈ (O‘(bday‘𝐵)) ∧ 𝐴 <s 𝐵))
23	20, 21, 22	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐴 ∈ (L‘𝐵))
24		fnfvima 7259	. . 3 ⊢ (( -s Fn No ∧ (L‘𝐵) ⊆ No ∧ 𝐴 ∈ (L‘𝐵)) → ( -s‘𝐴) ∈ ( -s “ (L‘𝐵)))
25	13, 14, 23, 24	mp3an12i 1495	. 2 ⊢ (𝜑 → ( -s‘𝐴) ∈ ( -s “ (L‘𝐵)))
26	9, 12, 25	sltssepcd 28185	1 ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∪ cun 4152  {csn 4588   class class class wbr 5106   “ cima 5658  Oncon0 6377   Fn wfn 6550  ‘cfv 6555  Nocsur 28024   <s clts 28025  bdaycbday 28026   <<s cslts 28170  Ocold 28236  Lcleft 28238  Rcright 28239   -_scnegs 28432

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-slts 28171  df-cuts 28173  df-0s 28220  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-negs 28434

This theorem is used by:  negsproplem7  28447