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Theorem addsproplem2 28144

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 6896	. . . . 5 ⊢ ( L ‘𝑋) ∈ V
2	1	abrexex 7960	. . . 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 6896	. . . . 5 ⊢ ( L ‘𝑌) ∈ V
5	4	abrexex 7960	. . . 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 7754	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ∈ V)
8		fvex 6896	. . . . 5 ⊢ ( R ‘𝑋) ∈ V
9	8	abrexex 7960	. . . 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 6896	. . . . 5 ⊢ ( R ‘𝑌) ∈ V
12	11	abrexex 7960	. . . 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 7754	. 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 485	. . . . . . . 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 28051	. . . . . . . . 9 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ No )
18	17	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑙 ∈ No )
19		addsproplem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
20	19	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 𝑌 ∈ No )
21		0no 27983	. . . . . . . . 9 ⊢ 0_s ∈ No
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → 0_s ∈ No )
23		bday0 27985	. . . . . . . . . . . . 13 ⊢ ( bday ‘ 0_s ) = ∅
24	23	oveq2i 7423	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = (( bday ‘𝑙) +no ∅)
25		bdayon 27926	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑙) ∈ On
26		naddrid 8671	. . . . . . . . . . . . 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 2786	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) +no ( bday ‘ 0_s )) = ( bday ‘𝑙)
29	28	uneq2i 4120	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙))
30		bdayon 27926	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑌) ∈ On
31		naddword1 8679	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)))
32	25, 30, 31	mp2an 704	. . . . . . . . . . 11 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌))
33		ssequn2 4143	. . . . . . . . . . 11 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
34	32, 33	mpbi 233	. . . . . . . . . 10 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
35	29, 34	eqtri 2786	. . . . . . . . 9 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑌))
36		leftold 28049	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 ∈ ( O ‘( bday ‘𝑋)))
37		bdayon 27926	. . . . . . . . . . . . . 14 ⊢ ( bday ‘𝑋) ∈ On
38		oldbday 28075	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑙 ∈ No ) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
39	37, 17, 38	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ ( L ‘𝑋) → (𝑙 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑙) ∈ ( bday ‘𝑋)))
40	36, 39	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → ( bday ‘𝑙) ∈ ( bday ‘𝑋))
41		naddel1 8675	. . . . . . . . . . . . 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 1490	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
43	40, 42	sylib 221	. . . . . . . . . . 11 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
44	43	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
45		elun1 4136	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
47	35, 46	eqeltrid 2867	. . . . . . . 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 28143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → ((𝑙 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑙) <s ( 0_s +_s 𝑙))))
49	48	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑙 +_s 𝑌) ∈ No )
50		eleq1a 2858	. . . . . 6 ⊢ ((𝑙 +_s 𝑌) ∈ No → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
51	49, 50	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝑋)) → (𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
52	51	rexlimdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) → 𝑝 ∈ No ))
53	52	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ⊆ No )
54	15	adantr 485	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑋 ∈ No )
57		leftno 28051	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ No )
58	57	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 𝑚 ∈ No )
59	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → 0_s ∈ No )
60	23	oveq2i 7423	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = (( bday ‘𝑋) +no ∅)
61		naddrid 8671	. . . . . . . . . . . . 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 2786	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) +no ( bday ‘ 0_s )) = ( bday ‘𝑋)
64	63	uneq2i 4120	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋))
65		bdayon 27926	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑚) ∈ On
66		naddword1 8679	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)))
67	37, 65, 66	mp2an 704	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚))
68		ssequn2 4143	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
69	67, 68	mpbi 233	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
70	64, 69	eqtri 2786	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
71		leftold 28049	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
72		oldbday 28075	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
73	30, 57, 72	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ( L ‘𝑌) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
74	71, 73	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ( L ‘𝑌) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
75		naddel2 8676	. . . . . . . . . . . . 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 1490	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
77	74, 76	sylib 221	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ( L ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
78	77	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
79		elun1 4136	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
81	70, 80	eqeltrid 2867	. . . . . . . 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 28143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → ((𝑋 +_s 𝑚) ∈ No ∧ (𝑚 <s 0_s → (𝑚 +_s 𝑋) <s ( 0_s +_s 𝑋))))
83	82	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑋 +_s 𝑚) ∈ No )
84		eleq1a 2858	. . . . . 6 ⊢ ((𝑋 +_s 𝑚) ∈ No → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
85	83, 84	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ( L ‘𝑌)) → (𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
86	85	rexlimdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) → 𝑞 ∈ No ))
87	86	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ⊆ No )
88	53, 87	unssd 4146	. 2 ⊢ (𝜑 → ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)}) ⊆ No )
89	15	adantr 485	. . . . . . . 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 28052	. . . . . . . . 9 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ No )
91	90	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑟 ∈ No )
92	19	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 𝑌 ∈ No )
93	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → 0_s ∈ No )
94	23	oveq2i 7423	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = (( bday ‘𝑟) +no ∅)
95		bdayon 27926	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑟) ∈ On
96		naddrid 8671	. . . . . . . . . . . . 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 2786	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) +no ( bday ‘ 0_s )) = ( bday ‘𝑟)
99	98	uneq2i 4120	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟))
100		naddword1 8679	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)))
101	95, 30, 100	mp2an 704	. . . . . . . . . . 11 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌))
102		ssequn2 4143	. . . . . . . . . . 11 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑌)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
103	101, 102	mpbi 233	. . . . . . . . . 10 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
104	99, 103	eqtri 2786	. . . . . . . . 9 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑌))
105		rightold 28050	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ ( O ‘( bday ‘𝑋)))
106		oldbday 28075	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑋) ∈ On ∧ 𝑟 ∈ No ) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
107	37, 90, 106	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ ( R ‘𝑋) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ↔ ( bday ‘𝑟) ∈ ( bday ‘𝑋)))
108	105, 107	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ( R ‘𝑋) → ( bday ‘𝑟) ∈ ( bday ‘𝑋))
109		naddel1 8675	. . . . . . . . . . . . 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 1490	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
111	108, 110	sylib 221	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
112	111	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
113		elun1 4136	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
115	104, 114	eqeltrid 2867	. . . . . . . 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 28143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → ((𝑟 +_s 𝑌) ∈ No ∧ (𝑌 <s 0_s → (𝑌 +_s 𝑟) <s ( 0_s +_s 𝑟))))
117	116	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑟 +_s 𝑌) ∈ No )
118		eleq1a 2858	. . . . . 6 ⊢ ((𝑟 +_s 𝑌) ∈ No → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
119	117, 118	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ( R ‘𝑋)) → (𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
120	119	rexlimdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) → 𝑤 ∈ No ))
121	120	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ⊆ No )
122	15	adantr 485	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑋 ∈ No )
124		rightno 28052	. . . . . . . . 9 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ No )
125	124	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 𝑠 ∈ No )
126	21	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → 0_s ∈ No )
127	63	uneq2i 4120	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋))
128		bdayon 27926	. . . . . . . . . . . 12 ⊢ ( bday ‘𝑠) ∈ On
129		naddword1 8679	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)))
130	37, 128, 129	mp2an 704	. . . . . . . . . . 11 ⊢ ( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠))
131		ssequn2 4143	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) ⊆ (( bday ‘𝑋) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
132	130, 131	mpbi 233	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
133	127, 132	eqtri 2786	. . . . . . . . 9 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0_s ))) = (( bday ‘𝑋) +no ( bday ‘𝑠))
134		rightold 28050	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑠 ∈ ( O ‘( bday ‘𝑌)))
135		oldbday 28075	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑠 ∈ No ) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
136	30, 124, 135	sylancr 598	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → (𝑠 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑠) ∈ ( bday ‘𝑌)))
137	134, 136	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ( R ‘𝑌) → ( bday ‘𝑠) ∈ ( bday ‘𝑌))
138		naddel2 8676	. . . . . . . . . . . . 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 1490	. . . . . . . . . . . 12 ⊢ (( bday ‘𝑠) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
140	137, 139	sylib 221	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ( R ‘𝑌) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
141	140	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
142		elun1 4136	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
144	133, 143	eqeltrid 2867	. . . . . . . 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 28143	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → ((𝑋 +_s 𝑠) ∈ No ∧ (𝑠 <s 0_s → (𝑠 +_s 𝑋) <s ( 0_s +_s 𝑋))))
146	145	simpld 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑋 +_s 𝑠) ∈ No )
147		eleq1a 2858	. . . . . 6 ⊢ ((𝑋 +_s 𝑠) ∈ No → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
148	146, 147	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ( R ‘𝑌)) → (𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
149	148	rexlimdva 3166	. . . 4 ⊢ (𝜑 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) → 𝑡 ∈ No ))
150	149	abssdv 4022	. . 3 ⊢ (𝜑 → {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ⊆ No )
151	121, 150	unssd 4146	. 2 ⊢ (𝜑 → ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)}) ⊆ No )
152		elun 4108	. . . . . . 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 3459	. . . . . . . . 9 ⊢ 𝑎 ∈ V
154		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑙 +_s 𝑌) ↔ 𝑎 = (𝑙 +_s 𝑌)))
155	154	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌) ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌)))
156	153, 155	elab 3639	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +_s 𝑌)} ↔ ∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌))
157		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑞 = 𝑎 → (𝑞 = (𝑋 +_s 𝑚) ↔ 𝑎 = (𝑋 +_s 𝑚)))
158	157	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑞 = 𝑎 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚) ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚)))
159	153, 158	elab 3639	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +_s 𝑚)} ↔ ∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚))
160	156, 159	orbi12i 927	. . . . . . 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 278	. . . . . 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 4108	. . . . . . 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 3459	. . . . . . . . 9 ⊢ 𝑏 ∈ V
164		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑤 = 𝑏 → (𝑤 = (𝑟 +_s 𝑌) ↔ 𝑏 = (𝑟 +_s 𝑌)))
165	164	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑤 = 𝑏 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌) ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)))
166	163, 165	elab 3639	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +_s 𝑌)} ↔ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌))
167		eqeq1 2767	. . . . . . . . . 10 ⊢ (𝑡 = 𝑏 → (𝑡 = (𝑋 +_s 𝑠) ↔ 𝑏 = (𝑋 +_s 𝑠)))
168	167	rexbidv 3189	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠) ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)))
169	163, 168	elab 3639	. . . . . . . 8 ⊢ (𝑏 ∈ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +_s 𝑠)} ↔ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠))
170	166, 169	orbi12i 927	. . . . . . 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 278	. . . . . 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 639	. . . . 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 1028	. . . . 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 278	. . . 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 3237	. . . . . . 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 28036	. . . . . . . . . . . 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 782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ ( L ‘𝑋))
179		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ ( R ‘𝑋))
180	177, 178, 179	sltssepcd 27946	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 <s 𝑟)
181	15	adantr 485	. . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
183	17	ad2antrl 740	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑙 ∈ No )
184	90	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
185		naddcom 8670	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌)))
186	30, 25, 185	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑌))
187	43	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
188	186, 187	eqeltrid 2867	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
189		naddcom 8670	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌)))
190	30, 95, 189	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑌))
191	111	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
192	190, 191	eqeltrid 2867	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
193		naddcl 8664	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On)
194	30, 25, 193	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑙)) ∈ On
195		naddcl 8664	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On)
196	30, 95, 195	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑌) +no ( bday ‘𝑟)) ∈ On
197		naddcl 8664	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On)
198	37, 30, 197	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On
199		onunel 6470	. . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 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 4136	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 28143	. . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 <s 𝑟 → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
206	180, 205	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌))
207		breq12 5115	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑟 +_s 𝑌)))
208	206, 207	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
209	208	rexlimdvva 3222	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
210	175, 209	biimtrrid 246	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
211		reeanv 3237	. . . . . . 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 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) ∈ No )
213	15	adantr 485	. . . . . . . . . . . 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 740	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 ∈ No )
215	124	ad2antll 741	. . . . . . . . . . . 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 4120	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙))
218		naddword1 8679	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)))
219	25, 128, 218	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠))
220		ssequn2 4143	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑙) ⊆ (( bday ‘𝑙) +no ( bday ‘𝑠)) ↔ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
221	219, 220	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
222	217, 221	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑙) +no ( bday ‘𝑠)) ∪ (( bday ‘𝑙) +no ( bday ‘ 0_s ))) = (( bday ‘𝑙) +no ( bday ‘𝑠))
223		naddel1 8675	. . . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
225	40, 224	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ ( L ‘𝑋) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
226	225	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑠)))
227	140	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
228		ontr1 6410	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
231		elun1 4136	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 2867	. . . . . . . . . . . 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 28143	. . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) ∈ No )
236	146	adantrl 728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) ∈ No )
237		rightgt 28028	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ ( R ‘𝑌) → 𝑌 <s 𝑠)
238	237	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 <s 𝑠)
239	19	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑌 ∈ No )
240	43	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
241		naddcl 8664	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On)
242	25, 30, 241	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑌)) ∈ On
243		naddcl 8664	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑙) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On)
244	25, 128, 243	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑙) +no ( bday ‘𝑠)) ∈ On
245		onunel 6470	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 <s 𝑠 → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙)))
252	238, 251	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑌 +_s 𝑙) <s (𝑠 +_s 𝑙))
253	214, 239	addscomd 28141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) = (𝑌 +_s 𝑙))
254	214, 215	addscomd 28141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) = (𝑠 +_s 𝑙))
255	252, 253, 254	3brtr4d 5144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑙 +_s 𝑠))
256		leftlt 28027	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ ( L ‘𝑋) → 𝑙 <s 𝑋)
257	256	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑙 <s 𝑋)
258	55	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
259		naddcom 8670	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠)))
260	128, 25, 259	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) = (( bday ‘𝑙) +no ( bday ‘𝑠))
261	260, 230	eqeltrid 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
262		naddcom 8670	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠)))
263	128, 37, 262	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑠))
264	263, 227	eqeltrid 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
265		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑙) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On)
266	128, 25, 265	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑙)) ∈ On
267		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑠) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On)
268	128, 37, 267	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑠) +no ( bday ‘𝑋)) ∈ On
269		onunel 6470	. . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 28143	. . . . . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 <s 𝑋 → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠)))
276	257, 275	mpd 16	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑠) <s (𝑋 +_s 𝑠))
277	212, 235, 236, 255, 276	ltstrd 27908	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠))
278		breq12 5115	. . . . . . . . 9 ⊢ ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑙 +_s 𝑌) <s (𝑋 +_s 𝑠)))
279	277, 278	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝑋) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
280	279	rexlimdvva 3222	. . . . . . 7 ⊢ (𝜑 → (∃𝑙 ∈ ( L ‘𝑋)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑙 +_s 𝑌) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
281	211, 280	biimtrrid 246	. . . . . 6 ⊢ (𝜑 → ((∃𝑙 ∈ ( L ‘𝑋)𝑎 = (𝑙 +_s 𝑌) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
282	210, 281	jaod 872	. . . . 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 3237	. . . . . . 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 485	. . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 ∈ No )
286	57	ad2antrl 740	. . . . . . . . . . . 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 740	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
289	288, 79	syl 18	. . . . . . . . . . . . 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 2867	. . . . . . . . . . . 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 28143	. . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) ∈ No )
293	90	ad2antll 741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ No )
294	98	uneq2i 4120	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟))
295		naddword1 8679	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)))
296	95, 65, 295	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚))
297		ssequn2 4143	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑟) ⊆ (( bday ‘𝑟) +no ( bday ‘𝑚)) ↔ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
298	296, 297	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
299	294, 298	eqtri 2786	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑟) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑟) +no ( bday ‘ 0_s ))) = (( bday ‘𝑟) +no ( bday ‘𝑚))
300		naddel1 8675	. . . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) ∈ ( bday ‘𝑋) ↔ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
302	108, 301	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ( R ‘𝑋) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
303	302	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑚)))
304		ontr1 6410	. . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
307		elun1 4136	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 2867	. . . . . . . . . . . 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 28143	. . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) ∈ No )
312	19	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑌 ∈ No )
313	111	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
314	313, 113	syl 18	. . . . . . . . . . . . 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 2867	. . . . . . . . . . . 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 28143	. . . . . . . . . . 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 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) ∈ No )
318		rightval 28024	. . . . . . . . . . . . . . . 16 ⊢ ( R ‘𝑋) = {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟}
319	318	eleq2i 2855	. . . . . . . . . . . . . . 15 ⊢ (𝑟 ∈ ( R ‘𝑋) ↔ 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
320	319	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝑟 ∈ ( R ‘𝑋) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
321	320	ad2antll 741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟})
322		rabid 3437	. . . . . . . . . . . . 13 ⊢ (𝑟 ∈ {𝑟 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑋 <s 𝑟} ↔ (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
323	321, 322	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑋 <s 𝑟))
324	323	simprd 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑋 <s 𝑟)
325		naddcom 8670	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚)))
326	65, 37, 325	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) = (( bday ‘𝑋) +no ( bday ‘𝑚))
327	326, 288	eqeltrid 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
328		naddcom 8670	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚)))
329	65, 95, 328	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) = (( bday ‘𝑟) +no ( bday ‘𝑚))
330	329, 306	eqeltrid 2867	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
331		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On)
332	65, 37, 331	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑋)) ∈ On
333		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑟) ∈ On) → (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On)
334	65, 95, 333	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑚) +no ( bday ‘𝑟)) ∈ On
335		onunel 6470	. . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 28143	. . . . . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 <s 𝑟 → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚)))
342	324, 341	mpd 16	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑚))
343		leftval 28023	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝑌) = {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌}
344	343	eleq2i 2855	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ( L ‘𝑌) ↔ 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
345	344	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ( L ‘𝑌) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
346	345	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌})
347		rabid 3437	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ {𝑚 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑚 <s 𝑌} ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
348	346, 347	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑚 <s 𝑌))
349	348	simprd 500	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → 𝑚 <s 𝑌)
350		naddcl 8664	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On)
351	95, 65, 350	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑚)) ∈ On
352		naddcl 8664	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑟) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On)
353	95, 30, 352	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑟) +no ( bday ‘𝑌)) ∈ On
354		onunel 6470	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 <s 𝑌 → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟)))
361	349, 360	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑚 +_s 𝑟) <s (𝑌 +_s 𝑟))
362	293, 286	addscomd 28141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) = (𝑚 +_s 𝑟))
363	293, 312	addscomd 28141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑌) = (𝑌 +_s 𝑟))
364	361, 362, 363	3brtr4d 5144	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑟 +_s 𝑚) <s (𝑟 +_s 𝑌))
365	292, 311, 317, 342, 364	ltstrd 27908	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌))
366		breq12 5115	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑟 +_s 𝑌)))
367	365, 366	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑟 ∈ ( R ‘𝑋))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
368	367	rexlimdvva 3222	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑟 ∈ ( R ‘𝑋)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
369	283, 368	biimtrrid 246	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑟 ∈ ( R ‘𝑋)𝑏 = (𝑟 +_s 𝑌)) → 𝑎 <s 𝑏))
370		reeanv 3237	. . . . . . 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 28036	. . . . . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ ( L ‘𝑌))
374		simprr 784	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ ( R ‘𝑌))
375	372, 373, 374	sltssepcd 27946	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 <s 𝑠)
376	15	adantr 485	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑋 ∈ No )
378	57	ad2antrl 740	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑚 ∈ No )
379	124	ad2antll 741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → 𝑠 ∈ No )
380	77	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
381	140	ad2antll 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
382		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑚) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On)
383	37, 65, 382	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ On
384		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑠) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On)
385	37, 128, 384	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑠)) ∈ On
386		onunel 6470	. . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 28143	. . . . . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 <s 𝑠 → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋)))
393	375, 392	mpd 16	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑚 +_s 𝑋) <s (𝑠 +_s 𝑋))
394	377, 378	addscomd 28141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) = (𝑚 +_s 𝑋))
395	377, 379	addscomd 28141	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑠) = (𝑠 +_s 𝑋))
396	393, 394, 395	3brtr4d 5144	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠))
397		breq12 5115	. . . . . . . . 9 ⊢ ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → (𝑎 <s 𝑏 ↔ (𝑋 +_s 𝑚) <s (𝑋 +_s 𝑠)))
398	396, 397	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ( L ‘𝑌) ∧ 𝑠 ∈ ( R ‘𝑌))) → ((𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
399	398	rexlimdvva 3222	. . . . . . 7 ⊢ (𝜑 → (∃𝑚 ∈ ( L ‘𝑌)∃𝑠 ∈ ( R ‘𝑌)(𝑎 = (𝑋 +_s 𝑚) ∧ 𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
400	370, 399	biimtrrid 246	. . . . . 6 ⊢ (𝜑 → ((∃𝑚 ∈ ( L ‘𝑌)𝑎 = (𝑋 +_s 𝑚) ∧ ∃𝑠 ∈ ( R ‘𝑌)𝑏 = (𝑋 +_s 𝑠)) → 𝑎 <s 𝑏))
401	369, 400	jaod 872	. . . . 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 872	. . . 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 245	. . 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 1134	. 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 27942	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 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 {cab 2741 ∀wral 3079 ∃wrex 3089 {crab 3416 Vcvv 3455 ∪ cun 3904 ⊆ wss 3906 ∅c0 4287 class class class wbr 5110 Oncon0 6362 ‘cfv 6538 (class class class)co 7412 +no cnadd 8652 No csur 27785 <s clts 27786 bday cbday 27787 <<s cslts 27931 0_s c0s 27979 O cold 27997 L cleft 27999 R cright 28000 +_s cadds 28133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-1o 8454 df-2o 8455 df-nadd 8653 df-no 27788 df-lts 27789 df-bday 27790 df-slts 27932 df-cuts 27934 df-0s 27981 df-made 28001 df-old 28002 df-left 28004 df-right 28005 df-norec2 28123 df-adds 28134

This theorem is referenced by: addsproplem3 28145

W3C validator