Theorem slerec 27310

Description: A comparison law for surreals considered as cuts of sets of surreals. Definition from [Conway] p. 4. Theorem 4 of [Alling] p. 186. Theorem 2.5 of [Gonshor] p. 9. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
slerec	⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Distinct variable groups:   𝐴,𝑎,𝑑   𝐵,𝑎,𝑑   𝐶,𝑎,𝑑   𝐷,𝑎,𝑑   𝑋,𝑎,𝑑   𝑌,𝑎,𝑑

Proof of Theorem slerec

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scutcl 27293	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
2	1	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) ∈ No )
3		scutcl 27293	. . . . . . . 8 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
4	3	ad3antlr 730	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 |s 𝐷) ∈ No )
5		ssltss2 27281	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐷 ⊆ No )
6	5	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐷 ⊆ No )
7	6	sselda 3982	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ No )
8		simplr 768	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
9		scutcut 27292	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
10	9	simp3d 1145	. . . . . . . . . . 11 ⊢ (𝐶 <<s 𝐷 → {(𝐶 |s 𝐷)} <<s 𝐷)
11	10	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → {(𝐶 |s 𝐷)} <<s 𝐷)
12		ssltsep 27282	. . . . . . . . . 10 ⊢ ({(𝐶 |s 𝐷)} <<s 𝐷 → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
14		ovex 7439	. . . . . . . . . 10 ⊢ (𝐶 |s 𝐷) ∈ V
15		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 |s 𝐷) <s 𝑑))
16	15	ralbidv 3178	. . . . . . . . . 10 ⊢ (𝑎 = (𝐶 |s 𝐷) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑))
17	14, 16	ralsn 4685	. . . . . . . . 9 ⊢ (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑)
18	13, 17	sylib 217	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐶 |s 𝐷) <s 𝑑)
19	18	r19.21bi 3249	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 |s 𝐷) <s 𝑑)
20	2, 4, 7, 8, 19	slelttrd 27254	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 |s 𝐵) <s 𝑑)
21	20	ralrimiva 3147	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
22		ssltss1 27280	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
23	22	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 ⊆ No )
24	23	adantr 482	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 ⊆ No )
25	24	sselda 3982	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
26	1	ad3antrrr 729	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 |s 𝐵) ∈ No )
27	3	ad3antlr 730	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐶 |s 𝐷) ∈ No )
28		scutcut 27292	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
29	28	simp2d 1144	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 <<s {(𝐴 |s 𝐵)})
30	29	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
31	30	adantr 482	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → 𝐴 <<s {(𝐴 |s 𝐵)})
32		ssltsep 27282	. . . . . . . . . 10 ⊢ (𝐴 <<s {(𝐴 |s 𝐵)} → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
34	33	r19.21bi 3249	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → ∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑)
35		ovex 7439	. . . . . . . . 9 ⊢ (𝐴 |s 𝐵) ∈ V
36		breq2 5152	. . . . . . . . 9 ⊢ (𝑑 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 |s 𝐵)))
37	35, 36	ralsn 4685	. . . . . . . 8 ⊢ (∀𝑑 ∈ {(𝐴 |s 𝐵)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 |s 𝐵))
38	34, 37	sylib 217	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐴 |s 𝐵))
39		simplr 768	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
40	25, 26, 27, 38, 39	sltletrd 27253	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐶 |s 𝐷))
41	40	ralrimiva 3147	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
42	21, 41	jca 513	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)) → (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))
43		bdayelon 27268	. . . . . . 7 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
44	43	onordi 6473	. . . . . 6 ⊢ Ord ( bday ‘(𝐴 |s 𝐵))
45		ordn2lp 6382	. . . . . 6 ⊢ (Ord ( bday ‘(𝐴 |s 𝐵)) → ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
46	44, 45	ax-mp 5	. . . . 5 ⊢ ¬ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
47	3	ad2antlr 726	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
48	1	adantr 482	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
49	48	adantr 482	. . . . . . 7 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
50		sltnle 27246	. . . . . . 7 ⊢ (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 |s 𝐵) ∈ No ) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
51	47, 49, 50	syl2anc 585	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) ↔ ¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
52	3	ad3antlr 730	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ∈ No )
53		ssltex1 27278	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
54	53	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 ∈ V)
55		snex 5431	. . . . . . . . . . 11 ⊢ {(𝐶 |s 𝐷)} ∈ V
56	54, 55	jctir 522	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V))
57	22	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 ⊆ No )
58	52	snssd 4812	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} ⊆ No )
59		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
60		breq2 5152	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝐶 |s 𝐷) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 |s 𝐷)))
61	14, 60	ralsn 4685	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 |s 𝐷))
62	61	ralbii 3094	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))
63	59, 62	sylibr 233	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)
64	57, 58, 63	3jca 1129	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 ⊆ No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑))
65		brsslt 27277	. . . . . . . . . 10 ⊢ (𝐴 <<s {(𝐶 |s 𝐷)} ↔ ((𝐴 ∈ V ∧ {(𝐶 |s 𝐷)} ∈ V) ∧ (𝐴 ⊆ No ∧ {(𝐶 |s 𝐷)} ⊆ No ∧ ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑎 <s 𝑑)))
66	56, 64, 65	sylanbrc 584	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐴 <<s {(𝐶 |s 𝐷)})
67		ssltex2 27279	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
68	67	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 ∈ V)
69	68, 55	jctil 521	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V))
70		ssltss2 27281	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
71	70	ad3antrrr 729	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐵 ⊆ No )
72	52	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) ∈ No )
73	48	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐴 |s 𝐵) ∈ No )
74	71	sselda 3982	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
75		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
76	28	simp3d 1145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 <<s 𝐵 → {(𝐴 |s 𝐵)} <<s 𝐵)
77	76	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐵)
78		ssltsep 27282	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐴 |s 𝐵)} <<s 𝐵 → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
79	77, 78	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
80		breq1 5151	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 |s 𝐵) <s 𝑏))
81	80	ralbidv 3178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴 |s 𝐵) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏))
82	35, 81	ralsn 4685	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏)
83	79, 82	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏 ∈ 𝐵 (𝐴 |s 𝐵) <s 𝑏)
84	83	r19.21bi 3249	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐴 |s 𝐵) <s 𝑏)
85	72, 73, 74, 75, 84	slttrd 27252	. . . . . . . . . . . . 13 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝐶 |s 𝐷) <s 𝑏)
86	85	ralrimiva 3147	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏)
87		breq1 5151	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐶 |s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 |s 𝐷) <s 𝑏))
88	87	ralbidv 3178	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐶 |s 𝐷) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏))
89	14, 88	ralsn 4685	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 |s 𝐷) <s 𝑏)
90	86, 89	sylibr 233	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
91	58, 71, 90	3jca 1129	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐶 |s 𝐷)} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏))
92		brsslt 27277	. . . . . . . . . 10 ⊢ ({(𝐶 |s 𝐷)} <<s 𝐵 ↔ (({(𝐶 |s 𝐷)} ∈ V ∧ 𝐵 ∈ V) ∧ ({(𝐶 |s 𝐷)} ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑎 ∈ {(𝐶 |s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)))
93	69, 91, 92	sylanbrc 584	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐶 |s 𝐷)} <<s 𝐵)
94		sltirr 27239	. . . . . . . . . . . . . 14 ⊢ ((𝐴 |s 𝐵) ∈ No → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
95	49, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵))
96		breq1 5151	. . . . . . . . . . . . . 14 ⊢ ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ((𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
97	96	notbid 318	. . . . . . . . . . . . 13 ⊢ ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → (¬ (𝐴 |s 𝐵) <s (𝐴 |s 𝐵) ↔ ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
98	95, 97	syl5ibcom 244	. . . . . . . . . . . 12 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐴 |s 𝐵) = (𝐶 |s 𝐷) → ¬ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)))
99	98	necon2ad 2956	. . . . . . . . . . 11 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)))
100	99	imp 408	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷))
101	100	necomd 2997	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵))
102		scutbdaylt 27309	. . . . . . . . 9 ⊢ (((𝐶 |s 𝐷) ∈ No ∧ (𝐴 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐵) ∧ (𝐶 |s 𝐷) ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
103	52, 66, 93, 101, 102	syl121anc 1376	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)))
104	1	ad3antrrr 729	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐴 |s 𝐵) ∈ No )
105		ssltex1 27278	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ∈ V)
106	105	ad3antlr 730	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 ∈ V)
107		snex 5431	. . . . . . . . . . 11 ⊢ {(𝐴 |s 𝐵)} ∈ V
108	106, 107	jctir 522	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V))
109		ssltss1 27280	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
110	109	ad3antlr 730	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 ⊆ No )
111	104	snssd 4812	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} ⊆ No )
112	110	sselda 3982	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
113	52	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐶 |s 𝐷) ∈ No )
114	48	ad3antrrr 729	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐴 |s 𝐵) ∈ No )
115	9	simp2d 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 <<s 𝐷 → 𝐶 <<s {(𝐶 |s 𝐷)})
116	115	ad3antlr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐶 |s 𝐷)})
117		ssltsep 27282	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 <<s {(𝐶 |s 𝐷)} → ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
118	116, 117	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
119	118	r19.21bi 3249	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → ∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑)
120		breq2 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝐶 |s 𝐷) → (𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 |s 𝐷)))
121	14, 120	ralsn 4685	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ {(𝐶 |s 𝐷)}𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 |s 𝐷))
122	119, 121	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 <s (𝐶 |s 𝐷))
123		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → (𝐶 |s 𝐷) <s (𝐴 |s 𝐵))
124	112, 113, 114, 122, 123	slttrd 27252	. . . . . . . . . . . . 13 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → 𝑐 <s (𝐴 |s 𝐵))
125		breq2 5152	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 |s 𝐵)))
126	35, 125	ralsn 4685	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 |s 𝐵))
127	124, 126	sylibr 233	. . . . . . . . . . . 12 ⊢ (((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) ∧ 𝑐 ∈ 𝐶) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
128	127	ralrimiva 3147	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)
129	110, 111, 128	3jca 1129	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (𝐶 ⊆ No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎))
130		brsslt 27277	. . . . . . . . . 10 ⊢ (𝐶 <<s {(𝐴 |s 𝐵)} ↔ ((𝐶 ∈ V ∧ {(𝐴 |s 𝐵)} ∈ V) ∧ (𝐶 ⊆ No ∧ {(𝐴 |s 𝐵)} ⊆ No ∧ ∀𝑐 ∈ 𝐶 ∀𝑎 ∈ {(𝐴 |s 𝐵)}𝑐 <s 𝑎)))
131	108, 129, 130	sylanbrc 584	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐶 <<s {(𝐴 |s 𝐵)})
132		ssltex2 27279	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐷 ∈ V)
133	132	ad3antlr 730	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 ∈ V)
134	133, 107	jctil 521	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V))
135	5	ad3antlr 730	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → 𝐷 ⊆ No )
136		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
137		breq1 5151	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 |s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
138	137	ralbidv 3178	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴 |s 𝐵) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑))
139	35, 138	ralsn 4685	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑)
140	136, 139	sylibr 233	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
141	111, 135, 140	3jca 1129	. . . . . . . . . 10 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ({(𝐴 |s 𝐵)} ⊆ No ∧ 𝐷 ⊆ No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑))
142		brsslt 27277	. . . . . . . . . 10 ⊢ ({(𝐴 |s 𝐵)} <<s 𝐷 ↔ (({(𝐴 |s 𝐵)} ∈ V ∧ 𝐷 ∈ V) ∧ ({(𝐴 |s 𝐵)} ⊆ No ∧ 𝐷 ⊆ No ∧ ∀𝑎 ∈ {(𝐴 |s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)))
143	134, 141, 142	sylanbrc 584	. . . . . . . . 9 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → {(𝐴 |s 𝐵)} <<s 𝐷)
144		scutbdaylt 27309	. . . . . . . . 9 ⊢ (((𝐴 |s 𝐵) ∈ No ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ (𝐴 |s 𝐵) ≠ (𝐶 |s 𝐷)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
145	104, 131, 143, 100, 144	syl121anc 1376	. . . . . . . 8 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))
146	103, 145	jca 513	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) ∧ (𝐶 |s 𝐷) <s (𝐴 |s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵))))
147	146	ex 414	. . . . . 6 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → ((𝐶 |s 𝐷) <s (𝐴 |s 𝐵) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
148	51, 147	sylbird 260	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (¬ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘(𝐶 |s 𝐷)) ∧ ( bday ‘(𝐶 |s 𝐷)) ∈ ( bday ‘(𝐴 |s 𝐵)))))
149	46, 148	mt3i 149	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷))
150	42, 149	impbida 800	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))))
151		breq12 5153	. . . 4 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷)))
152		breq1 5151	. . . . . 6 ⊢ (𝑋 = (𝐴 |s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 |s 𝐵) <s 𝑑))
153	152	ralbidv 3178	. . . . 5 ⊢ (𝑋 = (𝐴 |s 𝐵) → (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑))
154		breq2 5152	. . . . . 6 ⊢ (𝑌 = (𝐶 |s 𝐷) → (𝑎 <s 𝑌 ↔ 𝑎 <s (𝐶 |s 𝐷)))
155	154	ralbidv 3178	. . . . 5 ⊢ (𝑌 = (𝐶 |s 𝐷) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑌 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))
156	153, 155	bi2anan9 638	. . . 4 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷))))
157	151, 156	bibi12d 346	. . 3 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)) ↔ ((𝐴 |s 𝐵) ≤s (𝐶 |s 𝐷) ↔ (∀𝑑 ∈ 𝐷 (𝐴 |s 𝐵) <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 |s 𝐷)))))
158	150, 157	imbitrrid 245	. 2 ⊢ ((𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷)) → ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌))))
159	158	impcom 409	1 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  Vcvv 3475   ⊆ wss 3948  {csn 4628   class class class wbr 5148  Ord word 6361  ‘cfv 6541  (class class class)co 7406   No csur 27133   <s cslt 27134   bday cbday 27135   ≤s csle 27237   <<s csslt 27272   |s cscut 27274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1o 8463  df-2o 8464  df-no 27136  df-slt 27137  df-bday 27138  df-sle 27238  df-sslt 27273  df-scut 27275

This theorem is referenced by:  sltrec  27311