Theorem negsproplem7 27947

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		bdayelon 27695	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2	1	onordi 6448	. . 3 ⊢ Ord ( bday ‘𝐴)
3		bdayelon 27695	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
4	3	onordi 6448	. . 3 ⊢ Ord ( bday ‘𝐵)
5		ordtri3or 6367	. . 3 ⊢ ((Ord ( bday ‘𝐴) ∧ Ord ( bday ‘𝐵)) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
6	2, 4, 5	mp2an 692	. 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 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 ∈ No )
11		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐵 ∈ No )
13		negsproplem4.3	. . . . . 6 ⊢ (𝜑 → 𝐴 <s 𝐵)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 <s 𝐵)
15		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16	8, 10, 12, 14, 15	negsproplem4 27944	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
17	16	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
18	7	adantr 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 ∈ No )
20	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 ∈ No )
21	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 <s 𝐵)
22		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( bday ‘𝐴) = ( bday ‘𝐵))
23	18, 19, 20, 21, 22	negsproplem6 27946	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
24	23	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) = ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
25	7	adantr 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
27	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
28	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 <s 𝐵)
29		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
30	25, 26, 27, 28, 29	negsproplem5 27945	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
31	30	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
32	17, 24, 31	3jaod 1431	. 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 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∪ cun 3915   class class class wbr 5110  Ord word 6334  ‘cfv 6514   No csur 27558   <s cslt 27559   bday cbday 27560   -u_s cnegs 27932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-negs 27934

This theorem is referenced by:  negsprop  27948