Theorem negsproplem7 28128

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)) → (( -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)

Assertion

Ref	Expression
negsproplem7	⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘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 27846	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2	1	onordi 6460	. . 3 ⊢ Ord ( bday ‘𝐴)
3		bdayon 27846	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
4	3	onordi 6460	. . 3 ⊢ Ord ( bday ‘𝐵)
5		ordtri3or 6379	. . 3 ⊢ ((Ord ( bday ‘𝐴) ∧ Ord ( bday ‘𝐵)) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
6	2, 4, 5	mp2an 702	. 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)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
8	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
9		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No )
10	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 ∈ No )
11		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
12	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐵 ∈ No )
13		negsproplem4.3	. . . . . 6 ⊢ (𝜑 → 𝐴 <s 𝐵)
14	13	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 <s 𝐵)
15		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16	8, 10, 12, 14, 15	negsproplem4 28125	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
17	16	ex 416	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
18	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
19	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 ∈ No )
20	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 ∈ No )
21	13	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 <s 𝐵)
22		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( bday ‘𝐴) = ( bday ‘𝐵))
23	18, 19, 20, 21, 22	negsproplem6 28127	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
24	23	ex 416	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) = ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
25	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
26	9	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
27	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
28	13	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 <s 𝐵)
29		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
30	25, 26, 27, 28, 29	negsproplem5 28126	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
31	30	ex 416	. . 3 ⊢ (𝜑 → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
32	17, 24, 31	3jaod 1450	. 2 ⊢ (𝜑 → ((( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
33	6, 32	mpi 20	1 ⊢ (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1098   = wceq 1561   ∈ wcel 2143  ∀wral 3077   ∪ cun 3903   class class class wbr 5101  Ord word 6346  ‘cfv 6522   No csur 27705   <s clts 27706   bday cbday 27707   -u_s cnegs 28113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8438  df-2o 8439  df-no 27708  df-lts 27709  df-bday 27710  df-slts 27852  df-cuts 27854  df-0s 27901  df-made 27921  df-old 27922  df-left 27924  df-right 27925  df-norec 28032  df-negs 28115

This theorem is referenced by:  negsprop  28129