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Theorem lesrec 27869

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

Step	Hyp	Ref	Expression
1		cutscl 27852	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → (𝐴 \|s 𝐵) ∈ No )
2	1	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 \|s 𝐵) ∈ No )
3		cutscl 27852	. . . . . . . 8 ⊢ (𝐶 <<s 𝐷 → (𝐶 \|s 𝐷) ∈ No )
4	3	ad3antlr 741	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 \|s 𝐷) ∈ No )
5		sltsss2 27836	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐷 ⊆ No )
6	5	ad2antlr 737	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐷 ⊆ No )
7	6	sselda 3936	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ No )
8		simplr 778	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷))
9		cutcuts 27851	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → ((𝐶 \|s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 \|s 𝐷)} ∧ {(𝐶 \|s 𝐷)} <<s 𝐷))
10	9	simp3d 1156	. . . . . . . . . . 11 ⊢ (𝐶 <<s 𝐷 → {(𝐶 \|s 𝐷)} <<s 𝐷)
11	10	ad2antlr 737	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → {(𝐶 \|s 𝐷)} <<s 𝐷)
12		sltssep 27837	. . . . . . . . . 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 7425	. . . . . . . . . 10 ⊢ (𝐶 \|s 𝐷) ∈ V
15		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 \|s 𝐷) <s 𝑑))
16	15	ralbidv 3184	. . . . . . . . . 10 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 \|s 𝐷) <s 𝑑))
17	14, 16	ralsn 4639	. . . . . . . . 9 ⊢ (∀𝑎 ∈ {(𝐶 \|s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 \|s 𝐷) <s 𝑑)
18	13, 17	sylib 220	. . . . . . . 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 3253	. . . . . . 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	leltstrd 27806	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 \|s 𝐵) <s 𝑑)
21	20	ralrimiva 3153	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑)
22		sltsss1 27835	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
23	22	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 ⊆ No )
24	23	adantr 484	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐴 ⊆ No )
25	24	sselda 3936	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
26	1	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 \|s 𝐵) ∈ No )
27	3	ad3antlr 741	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐶 \|s 𝐷) ∈ No )
28		cutcuts 27851	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → ((𝐴 \|s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 \|s 𝐵)} ∧ {(𝐴 \|s 𝐵)} <<s 𝐵))
29	28	simp2d 1155	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 <<s {(𝐴 \|s 𝐵)})
30	29	adantr 484	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 \|s 𝐵)})
31	30	adantr 484	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐴 <<s {(𝐴 \|s 𝐵)})
32		sltssep 27837	. . . . . . . . . 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 3253	. . . . . . . 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 7425	. . . . . . . . 9 ⊢ (𝐴 \|s 𝐵) ∈ V
36		breq2 5103	. . . . . . . . 9 ⊢ (𝑑 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 \|s 𝐵)))
37	35, 36	ralsn 4639	. . . . . . . 8 ⊢ (∀𝑑 ∈ {(𝐴 \|s 𝐵)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 \|s 𝐵))
38	34, 37	sylib 220	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐴 \|s 𝐵))
39		simplr 778	. . . . . . 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	ltlestrd 27805	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐶 \|s 𝐷))
41	40	ralrimiva 3153	. . . . 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 519	. . . 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		bdayon 27822	. . . . . . 7 ⊢ ( bday ‘(𝐴 \|s 𝐵)) ∈ On
44	43	onordi 6455	. . . . . 6 ⊢ Ord ( bday ‘(𝐴 \|s 𝐵))
45		ordn2lp 6362	. . . . . 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 737	. . . . . . 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 484	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝐴 \|s 𝐵) ∈ No )
49	48	adantr 484	. . . . . . 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		ltnles 27794	. . . . . . 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 593	. . . . . 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 741	. . . . . . . . 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		sltsex1 27833	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
54	53	ad3antrrr 740	. . . . . . . . . . 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 5395	. . . . . . . . . . 11 ⊢ {(𝐶 \|s 𝐷)} ∈ V
56	54, 55	jctir 528	. . . . . . . . . 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 740	. . . . . . . . . . 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 4744	. . . . . . . . . . 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 787	. . . . . . . . . . . 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 5103	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 \|s 𝐷)))
61	14, 60	ralsn 4639	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 \|s 𝐷))
62	61	ralbii 3107	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 \|s 𝐷))
63	59, 62	sylibr 236	. . . . . . . . . . 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 1140	. . . . . . . . . 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		brslts 27832	. . . . . . . . . 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 592	. . . . . . . . 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		sltsex2 27834	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
68	67	ad3antrrr 740	. . . . . . . . . . 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 527	. . . . . . . . . 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		sltsss2 27836	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
71	70	ad3antrrr 740	. . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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 1156	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 <<s 𝐵 → {(𝐴 \|s 𝐵)} <<s 𝐵)
77	76	ad3antrrr 740	. . . . . . . . . . . . . . . . 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		sltssep 27837	. . . . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 \|s 𝐵) <s 𝑏))
81	80	ralbidv 3184	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 \|s 𝐵) <s 𝑏))
82	35, 81	ralsn 4639	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 \|s 𝐵) <s 𝑏)
83	79, 82	sylib 220	. . . . . . . . . . . . . . 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 3253	. . . . . . . . . . . . . 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	ltstrd 27804	. . . . . . . . . . . . 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 3153	. . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 \|s 𝐷) <s 𝑏))
88	87	ralbidv 3184	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 \|s 𝐷) <s 𝑏))
89	14, 88	ralsn 4639	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐶 \|s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 \|s 𝐷) <s 𝑏)
90	86, 89	sylibr 236	. . . . . . . . . . 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 1140	. . . . . . . . . 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		brslts 27832	. . . . . . . . . 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 592	. . . . . . . . 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		ltsirr 27787	. . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 14 ⊢ ((𝐴 \|s 𝐵) = (𝐶 \|s 𝐷) → ((𝐴 \|s 𝐵) <s (𝐴 \|s 𝐵) ↔ (𝐶 \|s 𝐷) <s (𝐴 \|s 𝐵)))
97	96	notbid 320	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 2971	. . . . . . . . . . 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 410	. . . . . . . . . 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 3011	. . . . . . . . 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		cutbdaylt 27868	. . . . . . . . 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 1393	. . . . . . . 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 740	. . . . . . . . 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		sltsex1 27833	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ∈ V)
106	105	ad3antlr 741	. . . . . . . . . . 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 5395	. . . . . . . . . . 11 ⊢ {(𝐴 \|s 𝐵)} ∈ V
108	106, 107	jctir 528	. . . . . . . . . 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		sltsss1 27835	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
110	109	ad3antlr 741	. . . . . . . . . . 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 4744	. . . . . . . . . . 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 3936	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . 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 1155	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 <<s 𝐷 → 𝐶 <<s {(𝐶 \|s 𝐷)})
116	115	ad3antlr 741	. . . . . . . . . . . . . . . . 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		sltssep 27837	. . . . . . . . . . . . . . . . 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 3253	. . . . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝐶 \|s 𝐷) → (𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 \|s 𝐷)))
121	14, 120	ralsn 4639	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 \|s 𝐷))
122	119, 121	sylib 220	. . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . 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	ltstrd 27804	. . . . . . . . . . . . 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 5103	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 \|s 𝐵)))
126	35, 125	ralsn 4639	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 \|s 𝐵))
127	124, 126	sylibr 236	. . . . . . . . . . . 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 3153	. . . . . . . . . . 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 1140	. . . . . . . . . 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		brslts 27832	. . . . . . . . . 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 592	. . . . . . . . 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		sltsex2 27834	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐷 ∈ V)
133	132	ad3antlr 741	. . . . . . . . . . 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 527	. . . . . . . . . 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 741	. . . . . . . . . . 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 786	. . . . . . . . . . . 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 5102	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 \|s 𝐵) <s 𝑑))
138	137	ralbidv 3184	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑))
139	35, 138	ralsn 4639	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑)
140	136, 139	sylibr 236	. . . . . . . . . . 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 1140	. . . . . . . . . 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		brslts 27832	. . . . . . . . . 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 592	. . . . . . . . 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		cutbdaylt 27868	. . . . . . . . 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 1393	. . . . . . . 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 519	. . . . . . 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 416	. . . . . 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 262	. . . . 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 810	. . 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 5104	. . . 4 ⊢ ((𝑋 = (𝐴 \|s 𝐵) ∧ 𝑌 = (𝐶 \|s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)))
152		breq1 5102	. . . . . 6 ⊢ (𝑋 = (𝐴 \|s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 \|s 𝐵) <s 𝑑))
153	152	ralbidv 3184	. . . . 5 ⊢ (𝑋 = (𝐴 \|s 𝐵) → (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑))
154		breq2 5103	. . . . . 6 ⊢ (𝑌 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑌 ↔ 𝑎 <s (𝐶 \|s 𝐷)))
155	154	ralbidv 3184	. . . . 5 ⊢ (𝑌 = (𝐶 \|s 𝐷) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑌 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 \|s 𝐷)))
156	153, 155	bi2anan9 647	. . . 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 347	. . 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 248	. 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 411	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)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 Vcvv 3453 ⊆ wss 3904 {csn 4581 class class class wbr 5099 Ord word 6341 ‘cfv 6517 (class class class)co 7392 No csur 27681 <s clts 27682 bday cbday 27683 ≤s cles 27785 <<s cslts 27827 |s ccuts 27829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-ord 6345 df-on 6346 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-1o 8432 df-2o 8433 df-no 27684 df-lts 27685 df-bday 27686 df-les 27786 df-slts 27828 df-cuts 27830

This theorem is referenced by: lesrecd 27870 twocut 28493

W3C validator