Theorem negsproplem7 28435

Description: Lemma for surreal negation. Show the second half of the inductive hypothesis unconditionally. (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)) → (( -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)

Assertion

Ref	Expression
negsproplem7	⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem negsproplem7

Step	Hyp	Ref	Expression
1		bdayon 28153	. . . 4 ⊢ (bday‘𝐴) ∈ On
2	1	onordi 6483	. . 3 ⊢ Ord (bday‘𝐴)
3		bdayon 28153	. . . 4 ⊢ (bday‘𝐵) ∈ On
4	3	onordi 6483	. . 3 ⊢ Ord (bday‘𝐵)
5		ordtri3or 6402	. . 3 ⊢ ((Ord (bday‘𝐴) ∧ Ord (bday‘𝐵)) → ((bday‘𝐴) ∈ (bday‘𝐵) ∨ (bday‘𝐴) = (bday‘𝐵) ∨ (bday‘𝐵) ∈ (bday‘𝐴)))
6	2, 4, 5	mp2an 705	. 2 ⊢ ((bday‘𝐴) ∈ (bday‘𝐵) ∨ (bday‘𝐴) = (bday‘𝐵) ∨ (bday‘𝐵) ∈ (bday‘𝐴))
7		negsproplem.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
9		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → 𝐴 ∈ No)
11		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → 𝐵 ∈ No)
13		negsproplem4.3	. . . . . 6 ⊢ (𝜑 → 𝐴 <s 𝐵)
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → 𝐴 <s 𝐵)
15		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → (bday‘𝐴) ∈ (bday‘𝐵))
16	8, 10, 12, 14, 15	negsproplem4 28432	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝐴) ∈ (bday‘𝐵)) → ( -s‘𝐵) <s ( -s‘𝐴))
17	16	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝐴) ∈ (bday‘𝐵) → ( -s‘𝐵) <s ( -s‘𝐴)))
18	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
19	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐴 ∈ No)
20	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐵 ∈ No)
21	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → 𝐴 <s 𝐵)
22		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → (bday‘𝐴) = (bday‘𝐵))
23	18, 19, 20, 21, 22	negsproplem6 28434	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝐴) = (bday‘𝐵)) → ( -s‘𝐵) <s ( -s‘𝐴))
24	23	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝐴) = (bday‘𝐵) → ( -s‘𝐵) <s ( -s‘𝐴)))
25	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
26	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → 𝐴 ∈ No)
27	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → 𝐵 ∈ No)
28	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → 𝐴 <s 𝐵)
29		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → (bday‘𝐵) ∈ (bday‘𝐴))
30	25, 26, 27, 28, 29	negsproplem5 28433	. . . 4 ⊢ ((𝜑 ∧ (bday‘𝐵) ∈ (bday‘𝐴)) → ( -s‘𝐵) <s ( -s‘𝐴))
31	30	ex 418	. . 3 ⊢ (𝜑 → ((bday‘𝐵) ∈ (bday‘𝐴) → ( -s‘𝐵) <s ( -s‘𝐴)))
32	17, 24, 31	3jaod 1456	. 2 ⊢ (𝜑 → (((bday‘𝐴) ∈ (bday‘𝐵) ∨ (bday‘𝐴) = (bday‘𝐵) ∨ (bday‘𝐵) ∈ (bday‘𝐴)) → ( -s‘𝐵) <s ( -s‘𝐴)))
33	6, 32	mpi 21	1 ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1569   ∈ wcel 2146  ∀wral 3080   ∪ cun 4152   class class class wbr 5106  Ord word 6368  ‘cfv 6547  Nocsur 28012   <s clts 28013  bdaycbday 28014   -_scnegs 28420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 6311  df-ord 6372  df-on 6373  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-1o 8490  df-2o 8491  df-no 28015  df-lts 28016  df-bday 28017  df-slts 28159  df-cuts 28161  df-0s 28208  df-made 28228  df-old 28229  df-left 28231  df-right 28232  df-norec 28339  df-negs 28422

This theorem is used by:  negsprop  28436