Theorem scutun12 27761

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 27752	. . . . . . 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 27759	. . . . 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 27760	. . . . 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 7388	. . . . . 6 ⊢ (𝐴 |s 𝐵) ∈ V
13	12	snnz 4730	. . . . 5 ⊢ {(𝐴 |s 𝐵)} ≠ ∅
14		sslttr 27758	. . . . 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 27751	. . 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 3442	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
20	19	elima 6021	. . . . . . . . 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 4587	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
22	21	breq2d 5107	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {𝑧}))
23	21	breq1d 5105	. . . . . . . . . . 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 3652	. . . . . . . . 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 27722	. . . . . . . . . . . . . 14 ⊢ bday Fn No
29		fnbrfvb 6881	. . . . . . . . . . . . . 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 27755	. . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐶)
37		sssslt1 27746	. . . . . . . . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐷)
42		sssslt2 27747	. . . . . . . . . . . . . . . . . . . 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 5107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑧}))
47	21	breq1d 5105	. . . . . . . . . . . . . . . . . . 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 3644	. . . . . . . . . . . . . . . . 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 4031	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
52		fnfvima 7176	. . . . . . . . . . . . . . . . 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 4915	. . . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . 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 3135	. . . . . . . 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 3125	. . . . . 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 4916	. . . . . 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 4587	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
73	72	breq2d 5107	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 ∪ 𝐶) <<s {𝑦} ↔ (𝐴 ∪ 𝐶) <<s {(𝐴 |s 𝐵)}))
74	72	breq1d 5105	. . . . . . . . . 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 3644	. . . . . . . 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 4031	. . . . . . . 8 ⊢ {𝑦 ∈ No ∣ ((𝐴 ∪ 𝐶) <<s {𝑦} ∧ {𝑦} <<s (𝐵 ∪ 𝐷))} ⊆ No
79		fnfvima 7176	. . . . . . . 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 4915	. . . . . 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 3949	. . . 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 27750	. . . . . 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 6840	. . . . . 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 7337	. . . . 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 2739	. 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 2771	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 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  ∃!wreu 3346  {crab 3397   ∪ cun 3897   ⊆ wss 3899  ∅c0 4284  {csn 4577  ∩ cint 4899   class class class wbr 5095   “ cima 5624   Fn wfn 6484  ‘cfv 6489  ℩crio 7311  (class class class)co 7355   No csur 27588   bday cbday 27590   <<s csslt 27730   |s cscut 27732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1o 8394  df-2o 8395  df-no 27591  df-slt 27592  df-bday 27593  df-sslt 27731  df-scut 27733

This theorem is referenced by: (None)