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Theorem addsproplem2 27900

Description: Lemma for surreal addition properties. When proving closure for operations defined using norec and norec2, it is a strictly stronger statement to say that the cut defined is actually a cut than it is to say that the operation is closed. We will often prove this stronger statement. Here, we do so for the cut involved in surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)

**Hypotheses**
Ref	Expression
addsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
addsproplem2.2	⊢ (𝜑 → 𝑋 ∈ No )
addsproplem2.3	⊢ (𝜑 → 𝑌 ∈ No )

**Assertion**
Ref	Expression
addsproplem2	⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))

Distinct variable groups: 𝑋,𝑞,𝑡 𝑋,𝑝,𝑤 𝑌,𝑝,𝑤 𝑌,𝑞,𝑡 𝑥,𝑍,𝑦,𝑧 𝜑,𝑝,𝑟,𝑤 𝑋,𝑙,𝑚,𝑟,𝑠,𝑥,𝑦,𝑧 𝑌,𝑙,𝑚,𝑟,𝑠,𝑥,𝑦,𝑧 𝜑,𝑙,𝑞,𝑚,𝑠 𝜑,𝑡,𝑟,𝑠 𝑝,𝑙,𝑞,𝑟 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧) 𝑍(𝑤,𝑡,𝑚,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem addsproplem2 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6839	. . . . 5 ⊢ ( L ‘𝑋) ∈ V
2	1	abrexex 7904	. . . 4 ⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∈ V
3	2	a1i 11	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∈ V)
4		fvex 6839	. . . . 5 ⊢ ( L ‘𝑌) ∈ V
5	4	abrexex 7904	. . . 4 ⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ∈ V
6	5	a1i 11	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ∈ V)
7	3, 6	unexd 7694	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∈ V)
8		fvex 6839	. . . . 5 ⊢ ( R ‘𝑋) ∈ V
9	8	abrexex 7904	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∈ V)
11		fvex 6839	. . . . 5 ⊢ ( R ‘𝑌) ∈ V
12	11	abrexex 7904	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ∈ V
13	12	a1i 11	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ∈ V)
14	10, 13	unexd 7694	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ∈ V)
15		addsproplem.1	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
17		leftssno 27813	. . . . . . . . . 10 ⊢ ( L ‘𝑋) ⊆ No
18	17	sseli 3933	. . . . . . . . 9 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ No )
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑙 ∈ No )
20		addsproplem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑌 ∈ No )
22		0sno 27758	. . . . . . . . 9 ⊢ 0_s ∈ No
23	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 0_s ∈ No )
24		bday0s 27760	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0_s ) = ∅
25	24	oveq2i 7364	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = (( bday ‘𝑙) +no ∅)
26		bdayelon 27704	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑙) ∈ On
27		naddrid 8608	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑙) ∈ On → (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙))
28	26, 27	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙)
29	25, 28	eqtri 2752	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = ( bday ‘𝑙)
30	29	uneq2i 4118	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙))
31		bdayelon 27704	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑌) ∈ On
32		naddword1 8616	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)))
33	26, 31, 32	mp2an 692	. . . . . . . . . . 11 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌))
34		ssequn2 4142	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
35	33, 34	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
36	30, 35	eqtri 2752	. . . . . . . . 9 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑌))
37		leftssold 27811	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
38	37	sseli 3933	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ ( O ‘( bday ‘𝑋)))
39		bdayelon 27704	. . . . . . . . . . . . . 14 ⊢ ( bday ‘𝑋) ∈ On
40		oldbday 27833	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑙 ∈ No ) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
41	39, 18, 40	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
42	38, 41	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → ( bday ‘𝑙) ∈ ( bday ‘𝑋))
43		naddel1 8612	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
44	26, 39, 31, 43	mp3an 1463	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
45	42, 44	sylib 218	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
47		elun1 4135	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
48	46, 47	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
49	36, 48	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
50	16, 19, 21, 23, 49	addsproplem1 27899	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑙) <s ( 0_s +_s 𝑙))))
51	50	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑙 +_s 𝑌) ∈ No )
52		eleq1a 2823	. . . . . 6 ⊢ ((𝑙 +_s 𝑌) ∈ No → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
53	51, 52	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
54	53	rexlimdva 3130	. . . 4 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
55	54	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ⊆ No )
56	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
57		addsproplem2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ No )
58	57	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑋 ∈ No )
59		leftssno 27813	. . . . . . . . . 10 ⊢ ( L ‘𝑌) ⊆ No
60	59	sseli 3933	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ No )
61	60	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑚 ∈ No )
62	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 0_s ∈ No )
63	24	oveq2i 7364	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = (( bday ‘𝑋) +no ∅)
64		naddrid 8608	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑋) ∈ On → (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋))
65	39, 64	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋)
66	63, 65	eqtri 2752	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = ( bday ‘𝑋)
67	66	uneq2i 4118	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋))
68		bdayelon 27704	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑚) ∈ On
69		naddword1 8616	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)))
70	39, 68, 69	mp2an 692	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚))
71		ssequn2 4142	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
72	70, 71	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
73	67, 72	eqtri 2752	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
74		leftssold 27811	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝑌) ⊆ ( O ‘( bday ‘𝑌))
75	74	sseli 3933	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
76		oldbday 27833	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
77	31, 60, 76	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
78	75, 77	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ( L ‘𝑌) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
79		naddel2 8613	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
80	68, 31, 39, 79	mp3an 1463	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
81	78, 80	sylib 218	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ( L ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
82	81	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
83		elun1 4135	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
84	82, 83	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
85	73, 84	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
86	56, 58, 61, 62, 85	addsproplem1 27899	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
87	86	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑋 +_s 𝑚) ∈ No )
88		eleq1a 2823	. . . . . 6 ⊢ ((𝑋 +_s 𝑚) ∈ No → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
89	87, 88	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
90	89	rexlimdva 3130	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
91	90	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ⊆ No )
92	55, 91	unssd 4145	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ⊆ No )
93	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
94		rightssno 27814	. . . . . . . . . 10 ⊢ ( R ‘𝑋) ⊆ No
95	94	sseli 3933	. . . . . . . . 9 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ No )
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑟 ∈ No )
97	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑌 ∈ No )
98	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 0_s ∈ No )
99	24	oveq2i 7364	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = (( bday ‘𝑟) +no ∅)
100		bdayelon 27704	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑟) ∈ On
101		naddrid 8608	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑟) ∈ On → (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟))
102	100, 101	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟)
103	99, 102	eqtri 2752	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = ( bday ‘𝑟)
104	103	uneq2i 4118	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟))
105		naddword1 8616	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)))
106	100, 31, 105	mp2an 692	. . . . . . . . . . 11 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌))
107		ssequn2 4142	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
108	106, 107	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
109	104, 108	eqtri 2752	. . . . . . . . 9 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑌))
110		rightssold 27812	. . . . . . . . . . . . . 14 ⊢ ( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
111	110	sseli 3933	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ ( O ‘( bday ‘𝑋)))
112		oldbday 27833	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑟 ∈ No ) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
113	39, 95, 112	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
114	111, 113	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ( R ‘𝑋) → ( bday ‘𝑟) ∈ ( bday ‘𝑋))
115		naddel1 8612	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
116	100, 39, 31, 115	mp3an 1463	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
117	114, 116	sylib 218	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
118	117	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
119		elun1 4135	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
120	118, 119	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
121	109, 120	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
122	93, 96, 97, 98, 121	addsproplem1 27899	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
123	122	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑟 +_s 𝑌) ∈ No )
124		eleq1a 2823	. . . . . 6 ⊢ ((𝑟 +_s 𝑌) ∈ No → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
125	123, 124	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
126	125	rexlimdva 3130	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
127	126	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ⊆ No )
128	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
129	57	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑋 ∈ No )
130		rightssno 27814	. . . . . . . . . 10 ⊢ ( R ‘𝑌) ⊆ No
131	130	sseli 3933	. . . . . . . . 9 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ No )
132	131	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑠 ∈ No )
133	22	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 0_s ∈ No )
134	66	uneq2i 4118	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋))
135		bdayelon 27704	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑠) ∈ On
136		naddword1 8616	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)))
137	39, 135, 136	mp2an 692	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠))
138		ssequn2 4142	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
139	137, 138	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
140	134, 139	eqtri 2752	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑠))
141		rightssold 27812	. . . . . . . . . . . . . 14 ⊢ ( R ‘𝑌) ⊆ ( O ‘( bday ‘𝑌))
142	141	sseli 3933	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ ( O ‘( bday ‘𝑌)))
143		oldbday 27833	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑠 ∈ No ) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
144	31, 131, 143	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
145	142, 144	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ( R ‘𝑌) → ( bday ‘𝑠) ∈ ( bday ‘𝑌))
146		naddel2 8613	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
147	135, 31, 39, 146	mp3an 1463	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
148	145, 147	sylib 218	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ( R ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
149	148	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
150		elun1 4135	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
151	149, 150	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
152	140, 151	eqeltrid 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
153	128, 129, 132, 133, 152	addsproplem1 27899	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((𝑋 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑋) <s ( 0_s +_s 𝑋))))
154	153	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑋 +_s 𝑠) ∈ No )
155		eleq1a 2823	. . . . . 6 ⊢ ((𝑋 +_s 𝑠) ∈ No → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
156	154, 155	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
157	156	rexlimdva 3130	. . . 4 ⊢ (𝜑 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
158	157	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ⊆ No )
159	127, 158	unssd 4145	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ⊆ No )
160		elun 4106	. . . . . . 7 ⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}))
161		vex 3442	. . . . . . . . 9 ⊢ 𝑎 ∈ V
162		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑙 +_s 𝑌) ↔ 𝑎 = (𝑙 +_s 𝑌)))
163	162	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌)))
164	161, 163	elab 3637	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌))
165		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑞 = 𝑎 → (𝑞 = (𝑋 +_s 𝑚) ↔ 𝑎 = (𝑋 +_s 𝑚)))
166	165	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑞 = 𝑎 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
167	161, 166	elab 3637	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚))
168	164, 167	orbi12i 914	. . . . . . 7 ⊢ ((𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∨ 𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
169	160, 168	bitri 275	. . . . . 6 ⊢ (𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
170		elun 4106	. . . . . . 7 ⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))
171		vex 3442	. . . . . . . . 9 ⊢ 𝑏 ∈ V
172		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +_s 𝑌) ↔ 𝑏 = (𝑟 +_s 𝑌)))
173	172	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
174	171, 173	elab 3637	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌))
175		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑋 +_s 𝑠) ↔ 𝑏 = (𝑋 +_s 𝑠)))
176	175	rexbidv 3153	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
177	171, 176	elab 3637	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))
178	174, 177	orbi12i 914	. . . . . . 7 ⊢ ((𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∨ 𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
179	170, 178	bitri 275	. . . . . 6 ⊢ (𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ↔ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
180	169, 179	anbi12i 628	. . . . 5 ⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) ↔ ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))))
181		anddi 1012	. . . . 5 ⊢ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∨ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)) ∧ (∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌) ∨ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))))
182	180, 181	bitri 275	. . . 4 ⊢ ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) ↔ (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))))
183		reeanv 3201	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
184		lltropt 27804	. . . . . . . . . . . 12 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
185	184	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ( L ‘𝑋) <<s ( R ‘𝑋))
186		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ ( L ‘𝑋))
187		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ ( R ‘𝑋))
188	185, 186, 187	ssltsepcd 27723	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 <s 𝑟)
189	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
190	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
191	18	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ No )
192	95	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
193		naddcom 8607	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
194	31, 26, 193	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
195	45	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
196	194, 195	eqeltrid 2832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
197		naddcom 8607	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
198	31, 100, 197	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
199	117	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
200	198, 199	eqeltrid 2832	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
201		naddcl 8602	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On)
202	31, 26, 201	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On
203		naddcl 8602	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On)
204	31, 100, 203	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On
205		naddcl 8602	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On)
206	39, 31, 205	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On
207		onunel 6418	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
208	202, 204, 206, 207	mp3an 1463	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
209	196, 200, 208	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
210		elun1 4135	. . . . . . . . . . . . 13 ⊢ (((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
211	209, 210	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
212	189, 190, 191, 192, 211	addsproplem1 27899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑌 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))))
213	212	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
214	188, 213	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))
215		breq12 5100	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
216	214, 215	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
217	216	rexlimdvva 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
218	183, 217	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
219		reeanv 3201	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) ↔ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
220	51	adantrr 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) ∈ No )
221	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
222	18	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 ∈ No )
223	131	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
224	22	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 0_s ∈ No )
225	29	uneq2i 4118	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙))
226		naddword1 8616	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)))
227	26, 135, 226	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠))
228		ssequn2 4142	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
229	227, 228	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
230	225, 229	eqtri 2752	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑠))
231		naddel1 8612	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠))))
232	26, 39, 135, 231	mp3an 1463	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
233	42, 232	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
234	233	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
235	148	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
236		ontr1 6358	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On → (((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
237	206, 236	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
238	234, 235, 237	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
239		elun1 4135	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
240	238, 239	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
241	230, 240	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
242	221, 222, 223, 224, 241	addsproplem1 27899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑙) <s ( 0_s +_s 𝑙))))
243	242	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) ∈ No )
244	154	adantrl 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) ∈ No )
245		rightgt 27796	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑌 <s 𝑠)
246	245	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 <s 𝑠)
247	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 ∈ No )
248	45	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
249		naddcl 8602	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On)
250	26, 31, 249	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On
251		naddcl 8602	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On)
252	26, 135, 251	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On
253		onunel 6418	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
254	250, 252, 206, 253	mp3an 1463	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
255	248, 238, 254	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
256		elun1 4135	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
257	255, 256	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
258	221, 222, 247, 223, 257	addsproplem1 27899	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))))
259	258	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙)))
260	246, 259	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))
261	222, 247	addscomd 27897	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) = (𝑌 +_s 𝑙))
262	222, 223	addscomd 27897	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) = (𝑠 +_s 𝑙))
263	260, 261, 262	3brtr4d 5127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑙 +_s 𝑠))
264		leftlt 27795	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 <s 𝑋)
265	264	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 <s 𝑋)
266	57	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
267		naddcom 8607	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
268	135, 26, 267	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
269	268, 238	eqeltrid 2832	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
270		naddcom 8607	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
271	135, 39, 270	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
272	271, 235	eqeltrid 2832	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
273		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On)
274	135, 26, 273	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On
275		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On)
276	135, 39, 275	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On
277		onunel 6418	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
278	274, 276, 206, 277	mp3an 1463	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
279	269, 272, 278	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
280		elun1 4135	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
281	279, 280	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
282	221, 223, 222, 266, 281	addsproplem1 27899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑠 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))))
283	282	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠)))
284	265, 283	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))
285	220, 243, 244, 263, 284	slttrd 27687	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠))
286		breq12 5100	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠)))
287	285, 286	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
288	287	rexlimdvva 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
289	219, 288	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
290	218, 289	jaod 859	. . . . 5 ⊢ (𝜑 → (((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) → 𝑎 <s 𝑏))
291		reeanv 3201	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
292	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
293	57	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 ∈ No )
294	60	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ No )
295	22	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 0_s ∈ No )
296	81	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
297	296, 83	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
298	73, 297	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
299	292, 293, 294, 295, 298	addsproplem1 27899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
300	299	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) ∈ No )
301	95	ad2antll 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
302	103	uneq2i 4118	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟))
303		naddword1 8616	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)))
304	100, 68, 303	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚))
305		ssequn2 4142	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
306	304, 305	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
307	302, 306	eqtri 2752	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑚))
308		naddel1 8612	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚))))
309	100, 39, 68, 308	mp3an 1463	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
310	114, 309	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
311	310	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
312		ontr1 6358	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On → (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
313	206, 312	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
314	311, 296, 313	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
315		elun1 4135	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
316	314, 315	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
317	307, 316	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
318	292, 301, 294, 295, 317	addsproplem1 27899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑟) <s ( 0_s +_s 𝑟))))
319	318	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) ∈ No )
320	20	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
321	117	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
322	321, 119	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
323	109, 322	eqeltrid 2832	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
324	292, 301, 320, 295, 323	addsproplem1 27899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
325	324	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) ∈ No )
326		rightval 27792	. . . . . . . . . . . . . . . 16 ⊢ ( R ‘𝑋) = {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟}
327	326	eleq2i 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ( R ‘𝑋) ↔ 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
328	327	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
329	328	ad2antll 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
330		rabid 3418	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟} ↔ (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
331	329, 330	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
332	331	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 <s 𝑟)
333		naddcom 8607	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
334	68, 39, 333	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
335	334, 296	eqeltrid 2832	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
336		naddcom 8607	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
337	68, 100, 336	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
338	337, 314	eqeltrid 2832	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
339		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On)
340	68, 39, 339	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On
341		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On)
342	68, 100, 341	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On
343		onunel 6418	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
344	340, 342, 206, 343	mp3an 1463	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
345	335, 338, 344	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
346		elun1 4135	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
347	345, 346	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
348	292, 294, 293, 301, 347	addsproplem1 27899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑚 +_s 𝑋) ∈ No ∧ (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))))
349	348	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚)))
350	332, 349	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))
351		leftval 27791	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝑌) = {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌}
352	351	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ( L ‘𝑌) ↔ 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
353	352	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
354	353	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
355		rabid 3418	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌} ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
356	354, 355	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
357	356	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 <s 𝑌)
358		naddcl 8602	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On)
359	100, 68, 358	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On
360		naddcl 8602	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On)
361	100, 31, 360	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On
362		onunel 6418	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
363	359, 361, 206, 362	mp3an 1463	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
364	314, 321, 363	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
365		elun1 4135	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
366	364, 365	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
367	292, 301, 294, 320, 366	addsproplem1 27899	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))))
368	367	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟)))
369	357, 368	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))
370	301, 294	addscomd 27897	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) = (𝑚 +_s 𝑟))
371	301, 320	addscomd 27897	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) = (𝑌 +_s 𝑟))
372	369, 370, 371	3brtr4d 5127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) <s (𝑟 +_s 𝑌))
373	300, 319, 325, 350, 372	slttrd 27687	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌))
374		breq12 5100	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌)))
375	373, 374	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
376	375	rexlimdvva 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
377	291, 376	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
378		reeanv 3201	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) ↔ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
379		lltropt 27804	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌)
380	379	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ( L ‘𝑌) <<s ( R ‘𝑌))
381		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ ( L ‘𝑌))
382		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ ( R ‘𝑌))
383	380, 381, 382	ssltsepcd 27723	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 <s 𝑠)
384	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +_s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +_s 𝑥) <s (𝑧 +_s 𝑥)))))
385	57	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
386	60	ad2antrl 728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ No )
387	131	ad2antll 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
388	81	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
389	148	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
390		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On)
391	39, 68, 390	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On
392		naddcl 8602	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On)
393	39, 135, 392	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On
394		onunel 6418	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On ∧ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On) → (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))))
395	391, 393, 206, 394	mp3an 1463	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∧ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
396	388, 389, 395	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
397		elun1 4135	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
398	396, 397	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
399	384, 385, 386, 387, 398	addsproplem1 27899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))))
400	399	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋)))
401	383, 400	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))
402	385, 386	addscomd 27897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) = (𝑚 +_s 𝑋))
403	385, 387	addscomd 27897	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) = (𝑠 +_s 𝑋))
404	401, 402, 403	3brtr4d 5127	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠))
405		breq12 5100	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠)))
406	404, 405	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
407	406	rexlimdvva 3186	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
408	378, 407	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
409	377, 408	jaod 859	. . . . 5 ⊢ (𝜑 → (((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) → 𝑎 <s 𝑏))
410	290, 409	jaod 859	. . . 4 ⊢ (𝜑 → ((((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))) ∨ ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) ∨ (∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))) → 𝑎 <s 𝑏))
411	182, 410	biimtrid 242	. . 3 ⊢ (𝜑 → ((𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) → 𝑎 <s 𝑏))
412	411	3impib 1116	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∧ 𝑏 ∈ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)})) → 𝑎 <s 𝑏)
413	7, 14, 92, 159, 412	ssltd 27720	1 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 {cab 2707 ∀wral 3044 ∃wrex 3053 {crab 3396 Vcvv 3438 ∪ cun 3903 ⊆ wss 3905 ∅c0 4286 class class class wbr 5095 Oncon0 6311 ‘cfv 6486 (class class class)co 7353 +no cnadd 8590 No csur 27567 <s cslt 27568 bday cbday 27569 <<s csslt 27709 0_s c0s 27754 O cold 27771 L cleft 27773 R cright 27774 +_s cadds 27889

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-1o 8395 df-2o 8396 df-nadd 8591 df-no 27570 df-slt 27571 df-bday 27572 df-sslt 27710 df-scut 27712 df-0s 27756 df-made 27775 df-old 27776 df-left 27778 df-right 27779 df-norec2 27879 df-adds 27890

This theorem is referenced by: addsproplem3 27901

W3C validator