Theorem negsproplem7 28420

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 28138	. . . 4 ⊢ (bday‘𝐴) ∈ On
2	1	onordi 6476	. . 3 ⊢ Ord (bday‘𝐴)
3		bdayon 28138	. . . 4 ⊢ (bday‘𝐵) ∈ On
4	3	onordi 6476	. . 3 ⊢ Ord (bday‘𝐵)
5		ordtri3or 6395	. . 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 28417	. . . 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 28419	. . . 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 28418	. . . 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 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897   class class class wbr 5103  Ord word 6361  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999   -_scnegs 28405

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  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 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-negs 28407

This theorem is used by:  negsprop  28421