Theorem scutun12 27779

Description: Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.)

Assertion

Ref	Expression
scutun12	⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))

Proof of Theorem scutun12

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s 𝐵)
2		scutcut 27770	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
3	1, 2	syl 17	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
4	3	simp2d 1143	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐴 <<s {(𝐴 |s 𝐵)})
5		simp2 1137	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → 𝐶 <<s {(𝐴 |s 𝐵)})
6		ssltun1 27777	. . . . 5 ⊢ ((𝐴 <<s {(𝐴 |s 𝐵)} ∧ 𝐶 <<s {(𝐴 |s 𝐵)}) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
7	4, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)})
8	3	simp3d 1144	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐵)
9		simp3 1138	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s 𝐷)
10		ssltun2 27778	. . . . 5 ⊢ (({(𝐴 |s 𝐵)} <<s 𝐵 ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))
12		ovex 7443	. . . . . 6 ⊢ (𝐴 |s 𝐵) ∈ V
13	12	snnz 4757	. . . . 5 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
14		sslttr 27776	. . . . 5 ⊢ (((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷) ∧ {(𝐴 |s 𝐵)} ≠ ∅) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
15	13, 14	mp3an3 1452	. . . 4 ⊢ (((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
16	7, 11, 15	syl2anc 584	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷))
17		scutval 27769	. . 3 ⊢ ((𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
18	16, 17	syl 17	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
19		vex 3468	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19	elima 6057	. . . . . . . . 9 ⊢ (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧 bday 𝑥)
21		sneq 4616	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
22	21	breq2d 5136	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {𝑧}))
23	21	breq1d 5134	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {𝑧} <<s (𝐵 ∪ 𝐷)))
24	22, 23	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷)) ↔ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))))
25	24	rexrab 3684	. . . . . . . . 9 ⊢ (∃𝑧 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}𝑧 bday 𝑥 ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥))
26	20, 25	bitri 275	. . . . . . . 8 ⊢ (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥))
27		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝑧 ∈ No )
28		bdayfn 27742	. . . . . . . . . . . . . 14 ⊢ bday Fn No
29		fnbrfvb 6934	. . . . . . . . . . . . . 14 ⊢ (( bday Fn No ∧ 𝑧 ∈ No ) → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
30	28, 29	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ No → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
31	27, 30	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (( bday ‘𝑧) = 𝑥 ↔ 𝑧 bday 𝑥))
32		simpll1 1213	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝐴 <<s 𝐵)
33		scutbday 27773	. . . . . . . . . . . . . . 15 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
34	32, 33	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
35		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝐴 ∪ 𝐶) <<s {𝑧})
36		ssun1 4158	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐶)
37		sssslt1 27764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ 𝐴 ⊆ (𝐴 ∪ 𝐶)) → 𝐴 <<s {𝑧})
38	36, 37	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∪ 𝐶) <<s {𝑧} → 𝐴 <<s {𝑧})
39	35, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝐴 <<s {𝑧})
40		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → {𝑧} <<s (𝐵 ∪ 𝐷))
41		ssun1 4158	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
42		sssslt2 27765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑧} <<s (𝐵 ∪ 𝐷) ∧ 𝐵 ⊆ (𝐵 ∪ 𝐷)) → {𝑧} <<s 𝐵)
43	41, 42	mpan2 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑧} <<s (𝐵 ∪ 𝐷) → {𝑧} <<s 𝐵)
44	40, 43	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → {𝑧} <<s 𝐵)
45	39, 44	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵))
46	21	breq2d 5136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
47	21	breq1d 5134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → ({𝑦} <<s 𝐵 ↔ {𝑧} <<s 𝐵))
48	46, 47	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
49	48	elrab 3676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑧 ∈ No ∧ (𝐴 <<s {𝑧} ∧ {𝑧} <<s 𝐵)))
50	27, 45, 49	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → 𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
51		ssrab2 4060	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52		fnfvima 7230	. . . . . . . . . . . . . . . . 17 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No ∧ 𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
53	28, 51, 52	mp3an12 1453	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
54	50, 53	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
55		intss1 4944	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑧) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑧))
56	54, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑧))
57	34, 56	eqsstrd 3998	. . . . . . . . . . . . 13 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧))
58		sseq2 3990	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) ↔ ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
59	58	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑧) = 𝑥 → (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
60	59	com12 32	. . . . . . . . . . . . 13 ⊢ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑧) → (( bday ‘𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
61	57, 60	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (( bday ‘𝑧) = 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
62	31, 61	sylbird 260	. . . . . . . . . . 11 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) ∧ ((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷))) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
63	62	ex 412	. . . . . . . . . 10 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) → (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) → (𝑧 bday 𝑥 → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)))
64	63	impd 410	. . . . . . . . 9 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) ∧ 𝑧 ∈ No ) → ((((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
65	64	rexlimdva 3142	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (∃𝑧 ∈ No (((𝐴 ∪ 𝐶) <<s {𝑧} ∧ {𝑧} <<s (𝐵 ∪ 𝐷)) ∧ 𝑧 bday 𝑥) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
66	26, 65	biimtrid 242	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥))
67	66	ralrimiv 3132	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∀𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
68		ssint 4945	. . . . . 6 ⊢ (( bday ‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ∀𝑥 ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑥)
69	67, 68	sylibr 234	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
70	3	simp1d 1142	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ No )
71	7, 11	jca 511	. . . . . . . 8 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)))
72		sneq 4616	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
73	72	breq2d 5136	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)}))
74	72	breq1d 5134	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s (𝐵 ∪ 𝐷) ↔ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷)))
75	73, 74	anbi12d 632	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 |s 𝐵) → (((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷)) ↔ ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))))
76	75	elrab 3676	. . . . . . . 8 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ↔ ((𝐴 |s 𝐵) ∈ No ∧ ((𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s (𝐵 ∪ 𝐷))))
77	70, 71, 76	sylanbrc 583	. . . . . . 7 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})
78		ssrab2 4060	. . . . . . . 8 ⊢ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No
79		fnfvima 7230	. . . . . . . 8 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No ∧ (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
80	28, 78, 79	mp3an12 1453	. . . . . . 7 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
81	77, 80	syl 17	. . . . . 6 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
82		intss1 4944	. . . . . 6 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) → ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
83	81, 82	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ⊆ ( bday ‘(𝐴 |s 𝐵)))
84	69, 83	eqssd 3981	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
85		conway 27768	. . . . . 6 ⊢ ((𝐴 ∪ 𝐶) <<s (𝐵 ∪ 𝐷) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
86	16, 85	syl 17	. . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}))
87		fveqeq2 6890	. . . . . 6 ⊢ (𝑥 = (𝐴 |s 𝐵) → (( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
88	87	riota2 7392	. . . . 5 ⊢ (((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ∧ ∃!𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵)))
89	77, 86, 88	syl2anc 584	. . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))}) ↔ (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵)))
90	84, 89	mpbid 232	. . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})) = (𝐴 |s 𝐵))
91	90	eqcomd 2742	. 2 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))})))
92	18, 91	eqtr4d 2774	1 ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  ∃!wreu 3362  {crab 3420   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ∩ cint 4927   class class class wbr 5124   “ cima 5662   Fn wfn 6531  ‘cfv 6536  ℩crio 7366  (class class class)co 7410   No csur 27608   bday cbday 27610   <<s csslt 27749   |s cscut 27751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1o 8485  df-2o 8486  df-no 27611  df-slt 27612  df-bday 27613  df-sslt 27750  df-scut 27752

This theorem is referenced by: (None)