Theorem negsproplem7 28040

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 27803	. . . 4 ⊢ ( bday ‘𝐴) ∈ On
2	1	onordi 6479	. . 3 ⊢ Ord ( bday ‘𝐴)
3		bdayelon 27803	. . . 4 ⊢ ( bday ‘𝐵) ∈ On
4	3	onordi 6479	. . 3 ⊢ Ord ( bday ‘𝐵)
5		ordtri3or 6400	. . 3 ⊢ ((Ord ( bday ‘𝐴) ∧ Ord ( bday ‘𝐵)) → (( bday ‘𝐴) ∈ ( bday ‘𝐵) ∨ ( bday ‘𝐴) = ( bday ‘𝐵) ∨ ( bday ‘𝐵) ∈ ( bday ‘𝐴)))
6	2, 4, 5	mp2an 690	. 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 479	. . . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 ∈ No )
11		negsproplem4.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ No )
12	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐵 ∈ No )
13		negsproplem4.3	. . . . . 6 ⊢ (𝜑 → 𝐴 <s 𝐵)
14	13	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → 𝐴 <s 𝐵)
15		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( bday ‘𝐴) ∈ ( bday ‘𝐵))
16	8, 10, 12, 14, 15	negsproplem4 28037	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) ∈ ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
17	16	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) ∈ ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
18	7	adantr 479	. . . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 ∈ No )
20	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐵 ∈ No )
21	13	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → 𝐴 <s 𝐵)
22		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( bday ‘𝐴) = ( bday ‘𝐵))
23	18, 19, 20, 21, 22	negsproplem6 28039	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
24	23	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝐴) = ( bday ‘𝐵) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
25	7	adantr 479	. . . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 ∈ No )
27	11	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐵 ∈ No )
28	13	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → 𝐴 <s 𝐵)
29		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( bday ‘𝐵) ∈ ( bday ‘𝐴))
30	25, 26, 27, 28, 29	negsproplem5 28038	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐵) ∈ ( bday ‘𝐴)) → ( -us ‘𝐵) <s ( -us ‘𝐴))
31	30	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝐵) ∈ ( bday ‘𝐴) → ( -us ‘𝐵) <s ( -us ‘𝐴)))
32	17, 24, 31	3jaod 1426	. 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 394   ∨ w3o 1083   = wceq 1534   ∈ wcel 2099  ∀wral 3051   ∪ cun 3944   class class class wbr 5145  Ord word 6367  ‘cfv 6546   No csur 27666   <s cslt 27667   bday cbday 27668   -u_s cnegs 28026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-1o 8488  df-2o 8489  df-no 27669  df-slt 27670  df-bday 27671  df-sslt 27808  df-scut 27810  df-0s 27851  df-made 27868  df-old 27869  df-left 27871  df-right 27872  df-norec 27949  df-negs 28028

This theorem is referenced by:  negsprop  28041