Theorem negsproplem1 27738

Description: Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-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)))))
negsproplem1.1	⊢ (𝜑 → 𝑋 ∈ No )
negsproplem1.2	⊢ (𝜑 → 𝑌 ∈ No )
negsproplem1.3	⊢ (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))

Assertion

Ref	Expression
negsproplem1	⊢ (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem negsproplem1

Step	Hyp	Ref	Expression
1		negsproplem1.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
2		negsproplem1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
3	1, 2	jca 511	. 2 ⊢ (𝜑 → (𝑋 ∈ No ∧ 𝑌 ∈ No ))
4		negsproplem.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
5		negsproplem1.3	. 2 ⊢ (𝜑 → (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)))
6		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
7	6	uneq1d 4163	. . . . 5 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ∪ ( bday ‘𝑦)) = (( bday ‘𝑋) ∪ ( bday ‘𝑦)))
8	7	eleq1d 2817	. . . 4 ⊢ (𝑥 = 𝑋 → ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑋) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
9		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( -us ‘𝑥) = ( -us ‘𝑋))
10	9	eleq1d 2817	. . . . 5 ⊢ (𝑥 = 𝑋 → (( -us ‘𝑥) ∈ No ↔ ( -us ‘𝑋) ∈ No ))
11		breq1 5152	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑦 ↔ 𝑋 <s 𝑦))
12	9	breq2d 5161	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( -us ‘𝑦) <s ( -us ‘𝑥) ↔ ( -us ‘𝑦) <s ( -us ‘𝑋)))
13	11, 12	imbi12d 343	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)) ↔ (𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋))))
14	10, 13	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝑋 → ((( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥))) ↔ (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋)))))
15	8, 14	imbi12d 343	. . 3 ⊢ (𝑥 = 𝑋 → (((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday ‘𝑋) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋))))))
16		fveq2 6892	. . . . . 6 ⊢ (𝑦 = 𝑌 → ( bday ‘𝑦) = ( bday ‘𝑌))
17	16	uneq2d 4164	. . . . 5 ⊢ (𝑦 = 𝑌 → (( bday ‘𝑋) ∪ ( bday ‘𝑦)) = (( bday ‘𝑋) ∪ ( bday ‘𝑌)))
18	17	eleq1d 2817	. . . 4 ⊢ (𝑦 = 𝑌 → ((( bday ‘𝑋) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) ↔ (( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵))))
19		breq2 5153	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 <s 𝑦 ↔ 𝑋 <s 𝑌))
20		fveq2 6892	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ( -us ‘𝑦) = ( -us ‘𝑌))
21	20	breq1d 5159	. . . . . 6 ⊢ (𝑦 = 𝑌 → (( -us ‘𝑦) <s ( -us ‘𝑋) ↔ ( -us ‘𝑌) <s ( -us ‘𝑋)))
22	19, 21	imbi12d 343	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋)) ↔ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))
23	22	anbi2d 628	. . . 4 ⊢ (𝑦 = 𝑌 → ((( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋))) ↔ (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋)))))
24	18, 23	imbi12d 343	. . 3 ⊢ (𝑦 = 𝑌 → (((( bday ‘𝑋) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑋)))) ↔ ((( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))))
25	15, 24	rspc2v 3623	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → (∀𝑥 ∈ No ∀𝑦 ∈ No ((( bday ‘𝑥) ∪ ( bday ‘𝑦)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) → ((( bday ‘𝑋) ∪ ( bday ‘𝑌)) ∈ (( bday ‘𝐴) ∪ ( bday ‘𝐵)) → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))))
26	3, 4, 5, 25	syl3c 66	1 ⊢ (𝜑 → (( -us ‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -us ‘𝑌) <s ( -us ‘𝑋))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060   ∪ cun 3947   class class class wbr 5149  ‘cfv 6544   No csur 27376   <s cslt 27377   bday cbday 27378   -u_s cnegs 27730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552

This theorem is referenced by:  negsproplem2  27739  negsproplem6  27743