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Theorem addsproplem2 27976

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 6847	. . . . 5 ⊢ ( L ‘𝑋) ∈ V
2	1	abrexex 7908	. . . 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 6847	. . . . 5 ⊢ ( L ‘𝑌) ∈ V
5	4	abrexex 7908	. . . 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 7701	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∈ V)
8		fvex 6847	. . . . 5 ⊢ ( R ‘𝑋) ∈ V
9	8	abrexex 7908	. . . 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 6847	. . . . 5 ⊢ ( R ‘𝑌) ∈ V
12	11	abrexex 7908	. . . 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 7701	. 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		leftno 27883	. . . . . . . . 9 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ No )
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑙 ∈ No )
19		addsproplem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑌 ∈ No )
21		0no 27815	. . . . . . . . 9 ⊢ 0_s ∈ No
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 0_s ∈ No )
23		bday0 27817	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0_s ) = ∅
24	23	oveq2i 7371	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = (( bday ‘𝑙) +no ∅)
25		bdayon 27758	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑙) ∈ On
26		naddrid 8612	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑙) ∈ On → (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙))
27	25, 26	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ∅) = ( bday ‘𝑙)
28	24, 27	eqtri 2760	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = ( bday ‘𝑙)
29	28	uneq2i 4106	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙))
30		bdayon 27758	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑌) ∈ On
31		naddword1 8620	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)))
32	25, 30, 31	mp2an 693	. . . . . . . . . . 11 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌))
33		ssequn2 4130	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
34	32, 33	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
35	29, 34	eqtri 2760	. . . . . . . . 9 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑌))
36		leftold 27881	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ ( O ‘( bday ‘𝑋)))
37		bdayon 27758	. . . . . . . . . . . . . 14 ⊢ ( bday ‘𝑋) ∈ On
38		oldbday 27907	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑙 ∈ No ) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
39	37, 17, 38	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
40	36, 39	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → ( bday ‘𝑙) ∈ ( bday ‘𝑋))
41		naddel1 8616	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
42	25, 37, 30, 41	mp3an 1464	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
43	40, 42	sylib 218	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
45		elun1 4123	. . . . . . . . . 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))))
46	44, 45	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
47	35, 46	eqeltrid 2841	. . . . . . . 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))))
48	16, 18, 20, 22, 47	addsproplem1 27975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑙) <s ( 0_s +_s 𝑙))))
49	48	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑙 +_s 𝑌) ∈ No )
50		eleq1a 2832	. . . . . 6 ⊢ ((𝑙 +_s 𝑌) ∈ No → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
51	49, 50	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
52	51	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
53	52	abssdv 4008	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ⊆ No )
54	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)))))
55		addsproplem2.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ No )
56	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑋 ∈ No )
57		leftno 27883	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ No )
58	57	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑚 ∈ No )
59	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 0_s ∈ No )
60	23	oveq2i 7371	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = (( bday ‘𝑋) +no ∅)
61		naddrid 8612	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑋) ∈ On → (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋))
62	37, 61	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ∅) = ( bday ‘𝑋)
63	60, 62	eqtri 2760	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = ( bday ‘𝑋)
64	63	uneq2i 4106	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋))
65		bdayon 27758	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑚) ∈ On
66		naddword1 8620	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)))
67	37, 65, 66	mp2an 693	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚))
68		ssequn2 4130	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
69	67, 68	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
70	64, 69	eqtri 2760	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
71		leftold 27881	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
72		oldbday 27907	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
73	30, 57, 72	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
74	71, 73	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ( L ‘𝑌) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
75		naddel2 8617	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
76	65, 30, 37, 75	mp3an 1464	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
77	74, 76	sylib 218	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ( L ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
78	77	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
79		elun1 4123	. . . . . . . . . 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))))
80	78, 79	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
81	70, 80	eqeltrid 2841	. . . . . . . 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))))
82	54, 56, 58, 59, 81	addsproplem1 27975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
83	82	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑋 +_s 𝑚) ∈ No )
84		eleq1a 2832	. . . . . 6 ⊢ ((𝑋 +_s 𝑚) ∈ No → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
85	83, 84	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
86	85	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
87	86	abssdv 4008	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ⊆ No )
88	53, 87	unssd 4133	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ⊆ No )
89	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)))))
90		rightno 27884	. . . . . . . . 9 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ No )
91	90	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑟 ∈ No )
92	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑌 ∈ No )
93	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 0_s ∈ No )
94	23	oveq2i 7371	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = (( bday ‘𝑟) +no ∅)
95		bdayon 27758	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑟) ∈ On
96		naddrid 8612	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝑟) ∈ On → (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟))
97	95, 96	ax-mp 5	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ∅) = ( bday ‘𝑟)
98	94, 97	eqtri 2760	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = ( bday ‘𝑟)
99	98	uneq2i 4106	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟))
100		naddword1 8620	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)))
101	95, 30, 100	mp2an 693	. . . . . . . . . . 11 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌))
102		ssequn2 4130	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
103	101, 102	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
104	99, 103	eqtri 2760	. . . . . . . . 9 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑌))
105		rightold 27882	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ ( O ‘( bday ‘𝑋)))
106		oldbday 27907	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑟 ∈ No ) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
107	37, 90, 106	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
108	105, 107	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ( R ‘𝑋) → ( bday ‘𝑟) ∈ ( bday ‘𝑋))
109		naddel1 8616	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
110	95, 37, 30, 109	mp3an 1464	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
111	108, 110	sylib 218	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
112	111	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
113		elun1 4123	. . . . . . . . . 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))))
114	112, 113	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
115	104, 114	eqeltrid 2841	. . . . . . . 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))))
116	89, 91, 92, 93, 115	addsproplem1 27975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
117	116	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑟 +_s 𝑌) ∈ No )
118		eleq1a 2832	. . . . . 6 ⊢ ((𝑟 +_s 𝑌) ∈ No → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
119	117, 118	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
120	119	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
121	120	abssdv 4008	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ⊆ No )
122	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)))))
123	55	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑋 ∈ No )
124		rightno 27884	. . . . . . . . 9 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ No )
125	124	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑠 ∈ No )
126	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 0_s ∈ No )
127	63	uneq2i 4106	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋))
128		bdayon 27758	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑠) ∈ On
129		naddword1 8620	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)))
130	37, 128, 129	mp2an 693	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠))
131		ssequn2 4130	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
132	130, 131	mpbi 230	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
133	127, 132	eqtri 2760	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑠))
134		rightold 27882	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ ( O ‘( bday ‘𝑌)))
135		oldbday 27907	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑠 ∈ No ) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
136	30, 124, 135	sylancr 588	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
137	134, 136	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ( R ‘𝑌) → ( bday ‘𝑠) ∈ ( bday ‘𝑌))
138		naddel2 8617	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
139	128, 30, 37, 138	mp3an 1464	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
140	137, 139	sylib 218	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ( R ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
141	140	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
142		elun1 4123	. . . . . . . . . 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))))
143	141, 142	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
144	133, 143	eqeltrid 2841	. . . . . . . 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))))
145	122, 123, 125, 126, 144	addsproplem1 27975	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((𝑋 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑋) <s ( 0_s +_s 𝑋))))
146	145	simpld 494	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑋 +_s 𝑠) ∈ No )
147		eleq1a 2832	. . . . . 6 ⊢ ((𝑋 +_s 𝑠) ∈ No → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
148	146, 147	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
149	148	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
150	149	abssdv 4008	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ⊆ No )
151	121, 150	unssd 4133	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ⊆ No )
152		elun 4094	. . . . . . 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)}))
153		vex 3434	. . . . . . . . 9 ⊢ 𝑎 ∈ V
154		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑙 +_s 𝑌) ↔ 𝑎 = (𝑙 +_s 𝑌)))
155	154	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌)))
156	153, 155	elab 3623	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌))
157		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑞 = 𝑎 → (𝑞 = (𝑋 +_s 𝑚) ↔ 𝑎 = (𝑋 +_s 𝑚)))
158	157	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑞 = 𝑎 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
159	153, 158	elab 3623	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚))
160	156, 159	orbi12i 915	. . . . . . 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)))
161	152, 160	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)))
162		elun 4094	. . . . . . 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)}))
163		vex 3434	. . . . . . . . 9 ⊢ 𝑏 ∈ V
164		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +_s 𝑌) ↔ 𝑏 = (𝑟 +_s 𝑌)))
165	164	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
166	163, 165	elab 3623	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌))
167		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑋 +_s 𝑠) ↔ 𝑏 = (𝑋 +_s 𝑠)))
168	167	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
169	163, 168	elab 3623	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))
170	166, 169	orbi12i 915	. . . . . . 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)))
171	162, 170	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)))
172	161, 171	anbi12i 629	. . . . 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))))
173		anddi 1013	. . . . 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)))))
174	172, 173	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)))))
175		reeanv 3210	. . . . . . 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)))
176		lltr 27868	. . . . . . . . . . . 12 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
177	176	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ( L ‘𝑋) <<s ( R ‘𝑋))
178		simprl 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ ( L ‘𝑋))
179		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ ( R ‘𝑋))
180	177, 178, 179	sltssepcd 27778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 <s 𝑟)
181	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)))))
182	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
183	17	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ No )
184	90	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
185		naddcom 8611	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
186	30, 25, 185	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
187	43	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
188	186, 187	eqeltrid 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
189		naddcom 8611	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
190	30, 95, 189	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
191	111	ad2antll 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
192	190, 191	eqeltrid 2841	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
193		naddcl 8606	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On)
194	30, 25, 193	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On
195		naddcl 8606	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On)
196	30, 95, 195	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On
197		naddcl 8606	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On)
198	37, 30, 197	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On
199		onunel 6424	. . . . . . . . . . . . . . 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)))))
200	194, 196, 198, 199	mp3an 1464	. . . . . . . . . . . . . 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))))
201	188, 192, 200	sylanbrc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑌) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑌) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
202		elun1 4123	. . . . . . . . . . . . 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))))
203	201, 202	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))))
204	181, 182, 183, 184, 203	addsproplem1 27975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑌 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))))
205	204	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
206	180, 205	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))
207		breq12 5091	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
208	206, 207	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
209	208	rexlimdvva 3195	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
210	175, 209	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
211		reeanv 3210	. . . . . . 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)))
212	49	adantrr 718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) ∈ No )
213	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)))))
214	17	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 ∈ No )
215	124	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
216	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 0_s ∈ No )
217	28	uneq2i 4106	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙))
218		naddword1 8620	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)))
219	25, 128, 218	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠))
220		ssequn2 4130	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
221	219, 220	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
222	217, 221	eqtri 2760	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑠))
223		naddel1 8616	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠))))
224	25, 37, 128, 223	mp3an 1464	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
225	40, 224	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
226	225	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
227	140	ad2antll 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
228		ontr1 6364	. . . . . . . . . . . . . . . 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))))
229	198, 228	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)))
230	226, 227, 229	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
231		elun1 4123	. . . . . . . . . . . . . 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))))
232	230, 231	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))))
233	222, 232	eqeltrid 2841	. . . . . . . . . . . 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))))
234	213, 214, 215, 216, 233	addsproplem1 27975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑙) <s ( 0_s +_s 𝑙))))
235	234	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) ∈ No )
236	146	adantrl 717	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) ∈ No )
237		rightgt 27860	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑌 <s 𝑠)
238	237	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 <s 𝑠)
239	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 ∈ No )
240	43	ad2antrl 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
241		naddcl 8606	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On)
242	25, 30, 241	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On
243		naddcl 8606	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On)
244	25, 128, 243	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On
245		onunel 6424	. . . . . . . . . . . . . . . . 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)))))
246	242, 244, 198, 245	mp3an 1464	. . . . . . . . . . . . . . . 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))))
247	240, 230, 246	sylanbrc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
248		elun1 4123	. . . . . . . . . . . . . . 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))))
249	247, 248	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))))
250	213, 214, 239, 215, 249	addsproplem1 27975	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))))
251	250	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙)))
252	238, 251	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))
253	214, 239	addscomd 27973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) = (𝑌 +_s 𝑙))
254	214, 215	addscomd 27973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) = (𝑠 +_s 𝑙))
255	252, 253, 254	3brtr4d 5118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑙 +_s 𝑠))
256		leftlt 27859	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 <s 𝑋)
257	256	ad2antrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 <s 𝑋)
258	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
259		naddcom 8611	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
260	128, 25, 259	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
261	260, 230	eqeltrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
262		naddcom 8611	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
263	128, 37, 262	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
264	263, 227	eqeltrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
265		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On)
266	128, 25, 265	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On
267		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On)
268	128, 37, 267	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On
269		onunel 6424	. . . . . . . . . . . . . . . 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)))))
270	266, 268, 198, 269	mp3an 1464	. . . . . . . . . . . . . . 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))))
271	261, 264, 270	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑠) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑠) +no ( bday ‘𝑋))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
272		elun1 4123	. . . . . . . . . . . . . 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))))
273	271, 272	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))))
274	213, 215, 214, 258, 273	addsproplem1 27975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑠 +_s 𝑙) ∈ No ∧ (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))))
275	274	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠)))
276	257, 275	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))
277	212, 235, 236, 255, 276	ltstrd 27741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠))
278		breq12 5091	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠)))
279	277, 278	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
280	279	rexlimdvva 3195	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
281	211, 280	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
282	210, 281	jaod 860	. . . . 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))
283		reeanv 3210	. . . . . . 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)))
284	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)))))
285	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 ∈ No )
286	57	ad2antrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ No )
287	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 0_s ∈ No )
288	77	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
289	288, 79	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))))
290	70, 289	eqeltrid 2841	. . . . . . . . . . . 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))))
291	284, 285, 286, 287, 290	addsproplem1 27975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
292	291	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) ∈ No )
293	90	ad2antll 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
294	98	uneq2i 4106	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟))
295		naddword1 8620	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)))
296	95, 65, 295	mp2an 693	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚))
297		ssequn2 4130	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
298	296, 297	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
299	294, 298	eqtri 2760	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑚))
300		naddel1 8616	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚))))
301	95, 37, 65, 300	mp3an 1464	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
302	108, 301	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
303	302	ad2antll 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
304		ontr1 6364	. . . . . . . . . . . . . . . 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))))
305	198, 304	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)))
306	303, 288, 305	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
307		elun1 4123	. . . . . . . . . . . . . 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))))
308	306, 307	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))))
309	299, 308	eqeltrid 2841	. . . . . . . . . . . 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))))
310	284, 293, 286, 287, 309	addsproplem1 27975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑟) <s ( 0_s +_s 𝑟))))
311	310	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) ∈ No )
312	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
313	111	ad2antll 730	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
314	313, 113	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))))
315	104, 314	eqeltrid 2841	. . . . . . . . . . . 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))))
316	284, 293, 312, 287, 315	addsproplem1 27975	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
317	316	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) ∈ No )
318		rightval 27856	. . . . . . . . . . . . . . . 16 ⊢ ( R ‘𝑋) = {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟}
319	318	eleq2i 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ( R ‘𝑋) ↔ 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
320	319	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
321	320	ad2antll 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
322		rabid 3411	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟} ↔ (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
323	321, 322	sylib 218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
324	323	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 <s 𝑟)
325		naddcom 8611	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
326	65, 37, 325	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
327	326, 288	eqeltrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
328		naddcom 8611	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
329	65, 95, 328	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
330	329, 306	eqeltrid 2841	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
331		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On)
332	65, 37, 331	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On
333		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On)
334	65, 95, 333	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On
335		onunel 6424	. . . . . . . . . . . . . . . 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)))))
336	332, 334, 198, 335	mp3an 1464	. . . . . . . . . . . . . . 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))))
337	327, 330, 336	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑚) +no ( bday ‘𝑋)) ∪ (( bday ‘𝑚) +no ( bday ‘𝑟))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
338		elun1 4123	. . . . . . . . . . . . . 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))))
339	337, 338	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))))
340	284, 286, 285, 293, 339	addsproplem1 27975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑚 +_s 𝑋) ∈ No ∧ (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))))
341	340	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚)))
342	324, 341	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))
343		leftval 27855	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝑌) = {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌}
344	343	eleq2i 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ( L ‘𝑌) ↔ 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
345	344	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
346	345	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
347		rabid 3411	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌} ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
348	346, 347	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
349	348	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 <s 𝑌)
350		naddcl 8606	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On)
351	95, 65, 350	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On
352		naddcl 8606	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On)
353	95, 30, 352	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On
354		onunel 6424	. . . . . . . . . . . . . . . . 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)))))
355	351, 353, 198, 354	mp3an 1464	. . . . . . . . . . . . . . . 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))))
356	306, 313, 355	sylanbrc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘𝑌))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
357		elun1 4123	. . . . . . . . . . . . . . 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))))
358	356, 357	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))))
359	284, 293, 286, 312, 358	addsproplem1 27975	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑟 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))))
360	359	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟)))
361	349, 360	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))
362	293, 286	addscomd 27973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) = (𝑚 +_s 𝑟))
363	293, 312	addscomd 27973	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) = (𝑌 +_s 𝑟))
364	361, 362, 363	3brtr4d 5118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) <s (𝑟 +_s 𝑌))
365	292, 311, 317, 342, 364	ltstrd 27741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌))
366		breq12 5091	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌)))
367	365, 366	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
368	367	rexlimdvva 3195	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
369	283, 368	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
370		reeanv 3210	. . . . . . 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)))
371		lltr 27868	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌)
372	371	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ( L ‘𝑌) <<s ( R ‘𝑌))
373		simprl 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ ( L ‘𝑌))
374		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ ( R ‘𝑌))
375	372, 373, 374	sltssepcd 27778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 <s 𝑠)
376	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)))))
377	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
378	57	ad2antrl 729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ No )
379	124	ad2antll 730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
380	77	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
381	140	ad2antll 730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
382		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On)
383	37, 65, 382	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On
384		naddcl 8606	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On)
385	37, 128, 384	mp2an 693	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On
386		onunel 6424	. . . . . . . . . . . . . . . 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)))))
387	383, 385, 198, 386	mp3an 1464	. . . . . . . . . . . . . . 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))))
388	380, 381, 387	sylanbrc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑠))) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
389		elun1 4123	. . . . . . . . . . . . . 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))))
390	388, 389	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))))
391	376, 377, 378, 379, 390	addsproplem1 27975	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))))
392	391	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋)))
393	375, 392	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))
394	377, 378	addscomd 27973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) = (𝑚 +_s 𝑋))
395	377, 379	addscomd 27973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) = (𝑠 +_s 𝑋))
396	393, 394, 395	3brtr4d 5118	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠))
397		breq12 5091	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠)))
398	396, 397	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
399	398	rexlimdvva 3195	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
400	370, 399	biimtrrid 243	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
401	369, 400	jaod 860	. . . . 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))
402	282, 401	jaod 860	. . . 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))
403	174, 402	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))
404	403	3impib 1117	. 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)
405	7, 14, 88, 151, 404	sltsd 27774	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 848 = wceq 1542 ∈ wcel 2114 {cab 2715 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∪ cun 3888 ⊆ wss 3890 ∅c0 4274 class class class wbr 5086 Oncon0 6317 ‘cfv 6492 (class class class)co 7360 +no cnadd 8594 No csur 27617 <s clts 27618 bday cbday 27619 <<s cslts 27763 0_s c0s 27811 O cold 27829 L cleft 27831 R cright 27832 +_s cadds 27965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-1o 8398 df-2o 8399 df-nadd 8595 df-no 27620 df-lts 27621 df-bday 27622 df-slts 27764 df-cuts 27766 df-0s 27813 df-made 27833 df-old 27834 df-left 27836 df-right 27837 df-norec2 27955 df-adds 27966

This theorem is referenced by: addsproplem3 27977

W3C validator