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Theorem lesrec 28167

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 28150	. . . . . . . 8 ⊢ (𝐴 <<s 𝐵 → (𝐴 \|s 𝐵) ∈ No )
2	1	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 \|s 𝐵) ∈ No )
3		cutscl 28150	. . . . . . . 8 ⊢ (𝐶 <<s 𝐷 → (𝐶 \|s 𝐷) ∈ No )
4	3	ad3antlr 744	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐶 \|s 𝐷) ∈ No )
5		sltsss2 28134	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐷 ⊆ No )
6	5	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐷 ⊆ No )
7	6	sselda 3931	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → 𝑑 ∈ No )
8		simplr 781	. . . . . . 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 28149	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → ((𝐶 \|s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 \|s 𝐷)} ∧ {(𝐶 \|s 𝐷)} <<s 𝐷))
10	9	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝐶 <<s 𝐷 → {(𝐶 \|s 𝐷)} <<s 𝐷)
11	10	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → {(𝐶 \|s 𝐷)} <<s 𝐷)
12		sltssep 28135	. . . . . . . . . 10 ⊢ ({(𝐶 \|s 𝐷)} <<s 𝐷 → ∀𝑎 ∈ {(𝐶 \|s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑)
13	11, 12	syl 18	. . . . . . . . 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 7445	. . . . . . . . . 10 ⊢ (𝐶 \|s 𝐷) ∈ V
15		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑑 ↔ (𝐶 \|s 𝐷) <s 𝑑))
16	15	ralbidv 3186	. . . . . . . . . 10 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 \|s 𝐷) <s 𝑑))
17	14, 16	ralsn 4642	. . . . . . . . 9 ⊢ (∀𝑎 ∈ {(𝐶 \|s 𝐷)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐶 \|s 𝐷) <s 𝑑)
18	13, 17	sylib 221	. . . . . . . 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 3255	. . . . . . 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 28104	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑑 ∈ 𝐷) → (𝐴 \|s 𝐵) <s 𝑑)
21	20	ralrimiva 3155	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑)
22		sltsss1 28133	. . . . . . . . . 10 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No )
23	22	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 ⊆ No )
24	23	adantr 486	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐴 ⊆ No )
25	24	sselda 3931	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ No )
26	1	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐴 \|s 𝐵) ∈ No )
27	3	ad3antlr 744	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → (𝐶 \|s 𝐷) ∈ No )
28		cutcuts 28149	. . . . . . . . . . . . 13 ⊢ (𝐴 <<s 𝐵 → ((𝐴 \|s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 \|s 𝐵)} ∧ {(𝐴 \|s 𝐵)} <<s 𝐵))
29	28	simp2d 1161	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 <<s {(𝐴 \|s 𝐵)})
30	29	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → 𝐴 <<s {(𝐴 \|s 𝐵)})
31	30	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) → 𝐴 <<s {(𝐴 \|s 𝐵)})
32		sltssep 28135	. . . . . . . . . 10 ⊢ (𝐴 <<s {(𝐴 \|s 𝐵)} → ∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐴 \|s 𝐵)}𝑎 <s 𝑑)
33	31, 32	syl 18	. . . . . . . . 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 3255	. . . . . . . 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 7445	. . . . . . . . 9 ⊢ (𝐴 \|s 𝐵) ∈ V
36		breq2 5107	. . . . . . . . 9 ⊢ (𝑑 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 \|s 𝐵)))
37	35, 36	ralsn 4642	. . . . . . . 8 ⊢ (∀𝑑 ∈ {(𝐴 \|s 𝐵)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐴 \|s 𝐵))
38	34, 37	sylib 221	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐴 \|s 𝐵))
39		simplr 781	. . . . . . 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 28103	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)) ∧ 𝑎 ∈ 𝐴) → 𝑎 <s (𝐶 \|s 𝐷))
41	40	ralrimiva 3155	. . . . 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 521	. . . 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 28120	. . . . . . 7 ⊢ ( bday ‘(𝐴 \|s 𝐵)) ∈ On
44	43	onordi 6469	. . . . . 6 ⊢ Ord ( bday ‘(𝐴 \|s 𝐵))
45		ordn2lp 6375	. . . . . 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 740	. . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) → (𝐴 \|s 𝐵) ∈ No )
49	48	adantr 486	. . . . . . 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 28092	. . . . . . 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 596	. . . . . 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 744	. . . . . . . . 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 28131	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
54	53	ad3antrrr 743	. . . . . . . . . . 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 5397	. . . . . . . . . . 11 ⊢ {(𝐶 \|s 𝐷)} ∈ V
56	54, 55	jctir 530	. . . . . . . . . 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 743	. . . . . . . . . . 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 4747	. . . . . . . . . . 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 790	. . . . . . . . . . . 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 5107	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 \|s 𝐷)))
61	14, 60	ralsn 4642	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑎 <s 𝑑 ↔ 𝑎 <s (𝐶 \|s 𝐷))
62	61	ralbii 3109	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑎 <s 𝑑 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 \|s 𝐷))
63	59, 62	sylibr 237	. . . . . . . . . . 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 1146	. . . . . . . . . 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 28130	. . . . . . . . . 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 595	. . . . . . . . 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 28132	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
68	67	ad3antrrr 743	. . . . . . . . . . 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 529	. . . . . . . . . 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 28134	. . . . . . . . . . . 12 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
71	70	ad3antrrr 743	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 <<s 𝐵 → {(𝐴 \|s 𝐵)} <<s 𝐵)
77	76	ad3antrrr 743	. . . . . . . . . . . . . . . . 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 28135	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐴 \|s 𝐵)} <<s 𝐵 → ∀𝑎 ∈ {(𝐴 \|s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏)
79	77, 78	syl 18	. . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑏 ↔ (𝐴 \|s 𝐵) <s 𝑏))
81	80	ralbidv 3186	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 \|s 𝐵) <s 𝑏))
82	35, 81	ralsn 4642	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐴 \|s 𝐵) <s 𝑏)
83	79, 82	sylib 221	. . . . . . . . . . . . . . 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 3255	. . . . . . . . . . . . . 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 28102	. . . . . . . . . . . . 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 3155	. . . . . . . . . . . 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 5106	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑏 ↔ (𝐶 \|s 𝐷) <s 𝑏))
88	87	ralbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐶 \|s 𝐷) → (∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 \|s 𝐷) <s 𝑏))
89	14, 88	ralsn 4642	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐶 \|s 𝐷)}∀𝑏 ∈ 𝐵 𝑎 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 (𝐶 \|s 𝐷) <s 𝑏)
90	86, 89	sylibr 237	. . . . . . . . . . 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 1146	. . . . . . . . . 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 28130	. . . . . . . . . 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 595	. . . . . . . . 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 28085	. . . . . . . . . . . . . 14 ⊢ ((𝐴 \|s 𝐵) ∈ No → ¬ (𝐴 \|s 𝐵) <s (𝐴 \|s 𝐵))
95	49, 94	syl 18	. . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . 14 ⊢ ((𝐴 \|s 𝐵) = (𝐶 \|s 𝐷) → ((𝐴 \|s 𝐵) <s (𝐴 \|s 𝐵) ↔ (𝐶 \|s 𝐷) <s (𝐴 \|s 𝐵)))
97	96	notbid 321	. . . . . . . . . . . . 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 248	. . . . . . . . . . . 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 412	. . . . . . . . . 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 28166	. . . . . . . . 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 1402	. . . . . . . 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 743	. . . . . . . . 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 28131	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ∈ V)
106	105	ad3antlr 744	. . . . . . . . . . 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 5397	. . . . . . . . . . 11 ⊢ {(𝐴 \|s 𝐵)} ∈ V
108	106, 107	jctir 530	. . . . . . . . . 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 28133	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
110	109	ad3antlr 744	. . . . . . . . . . 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 4747	. . . . . . . . . . 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 3931	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 <<s 𝐷 → 𝐶 <<s {(𝐶 \|s 𝐷)})
116	115	ad3antlr 744	. . . . . . . . . . . . . . . . 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 28135	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 <<s {(𝐶 \|s 𝐷)} → ∀𝑐 ∈ 𝐶 ∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑐 <s 𝑑)
118	116, 117	syl 18	. . . . . . . . . . . . . . . 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 3255	. . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝐶 \|s 𝐷) → (𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 \|s 𝐷)))
121	14, 120	ralsn 4642	. . . . . . . . . . . . . . 15 ⊢ (∀𝑑 ∈ {(𝐶 \|s 𝐷)}𝑐 <s 𝑑 ↔ 𝑐 <s (𝐶 \|s 𝐷))
122	119, 121	sylib 221	. . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 28102	. . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 \|s 𝐵)))
126	35, 125	ralsn 4642	. . . . . . . . . . . . 13 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}𝑐 <s 𝑎 ↔ 𝑐 <s (𝐴 \|s 𝐵))
127	124, 126	sylibr 237	. . . . . . . . . . . 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 3155	. . . . . . . . . . 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 1146	. . . . . . . . . 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 28130	. . . . . . . . . 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 595	. . . . . . . . 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 28132	. . . . . . . . . . . 12 ⊢ (𝐶 <<s 𝐷 → 𝐷 ∈ V)
133	132	ad3antlr 744	. . . . . . . . . . 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 529	. . . . . . . . . 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 744	. . . . . . . . . . 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 789	. . . . . . . . . . . 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 5106	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (𝑎 <s 𝑑 ↔ (𝐴 \|s 𝐵) <s 𝑑))
138	137	ralbidv 3186	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝐴 \|s 𝐵) → (∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑))
139	35, 138	ralsn 4642	. . . . . . . . . . . 12 ⊢ (∀𝑎 ∈ {(𝐴 \|s 𝐵)}∀𝑑 ∈ 𝐷 𝑎 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑)
140	136, 139	sylibr 237	. . . . . . . . . . 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 1146	. . . . . . . . . 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 28130	. . . . . . . . . 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 595	. . . . . . . . 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 28166	. . . . . . . . 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 1402	. . . . . . . 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 521	. . . . . . 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 418	. . . . . 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 263	. . . . 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 150	. . . 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 813	. . 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 5108	. . . 4 ⊢ ((𝑋 = (𝐴 \|s 𝐵) ∧ 𝑌 = (𝐶 \|s 𝐷)) → (𝑋 ≤s 𝑌 ↔ (𝐴 \|s 𝐵) ≤s (𝐶 \|s 𝐷)))
152		breq1 5106	. . . . . 6 ⊢ (𝑋 = (𝐴 \|s 𝐵) → (𝑋 <s 𝑑 ↔ (𝐴 \|s 𝐵) <s 𝑑))
153	152	ralbidv 3186	. . . . 5 ⊢ (𝑋 = (𝐴 \|s 𝐵) → (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ↔ ∀𝑑 ∈ 𝐷 (𝐴 \|s 𝐵) <s 𝑑))
154		breq2 5107	. . . . . 6 ⊢ (𝑌 = (𝐶 \|s 𝐷) → (𝑎 <s 𝑌 ↔ 𝑎 <s (𝐶 \|s 𝐷)))
155	154	ralbidv 3186	. . . . 5 ⊢ (𝑌 = (𝐶 \|s 𝐷) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑌 ↔ ∀𝑎 ∈ 𝐴 𝑎 <s (𝐶 \|s 𝐷)))
156	153, 155	bi2anan9 650	. . . 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 348	. . 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 249	. 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 413	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 Vcvv 3451 ⊆ wss 3899 {csn 4584 class class class wbr 5103 Ord word 6354 ‘cfv 6531 (class class class)co 7412 No csur 27979 <s clts 27980 bday cbday 27981 ≤s cles 28083 <<s cslts 28125 |s ccuts 28127

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-ord 6358 df-on 6359 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1o 8460 df-2o 8461 df-no 27982 df-lts 27983 df-bday 27984 df-les 28084 df-slts 28126 df-cuts 28128

This theorem is used by: lesrecd 28168 twocut 28791

W3C validator