Theorem cutbdaylt 27972

Description: If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
cutbdaylt	⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋))

Proof of Theorem cutbdaylt

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

Step	Hyp	Ref	Expression
1		simp2l 1218	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s {𝑋})
2		simp2r 1219	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} <<s 𝐵)
3		snnzg 4741	. . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} ≠ ∅)
4	3	3ad2ant1 1151	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} ≠ ∅)
5		sltstr 27961	. . . . 5 ⊢ ((𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵 ∧ {𝑋} ≠ ∅) → 𝐴 <<s 𝐵)
6	1, 2, 4, 5	syl3anc 1398	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
7		cutbday 27958	. . . 4 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
8	6, 7	syl 18	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
9		bdayfn 27922	. . . . 5 ⊢ bday Fn No
10		ssrab2 4035	. . . . 5 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
11		simp1 1154	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ No )
12		simp2 1155	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵))
13		sneq 4600	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → {𝑦} = {𝑋})
14	13	breq2d 5122	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑋}))
15	13	breq1d 5120	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ({𝑦} <<s 𝐵 ↔ {𝑋} <<s 𝐵))
16	14, 15	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
17	16	elrab 3651	. . . . . 6 ⊢ (𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
18	11, 12, 17	sylanbrc 594	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
19		fnfvima 7233	. . . . 5 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No ∧ 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ( bday ‘𝑋) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
20	9, 10, 18, 19	mp3an12i 1494	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘𝑋) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
21		intss1 4929	. . . 4 ⊢ (( bday ‘𝑋) ∈ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑋))
22	20, 21	syl 18	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ⊆ ( bday ‘𝑋))
23	8, 22	eqsstrd 3972	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋))
24		simprl 782	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s {𝑋})
25		simprr 784	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} <<s 𝐵)
26	3	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} ≠ ∅)
27	24, 25, 26, 5	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s 𝐵)
28	27, 7	syl 18	. . . . . . . . . 10 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
29	28	eqeq1d 2765	. . . . . . . . 9 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋) ↔ ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) = ( bday ‘𝑋)))
30		eqcom 2770	. . . . . . . . 9 ⊢ (∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) = ( bday ‘𝑋) ↔ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
31	29, 30	bitrdi 290	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋) ↔ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
32	31	biimpa 481	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
33	17	biimpri 231	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
34	27	adantr 485	. . . . . . . . 9 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → 𝐴 <<s 𝐵)
35		conway 27953	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
36	34, 35	syl 18	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
37		fveqeq2 6892	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
38	37	riota2 7394	. . . . . . . . 9 ⊢ ((𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ∧ ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) → (( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) = 𝑋))
39		eqcom 2770	. . . . . . . . 9 ⊢ ((℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) = 𝑋 ↔ 𝑋 = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
40	38, 39	bitrdi 290	. . . . . . . 8 ⊢ ((𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ∧ ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) → (( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ 𝑋 = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))))
41	33, 36, 40	syl2an2r 697	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → (( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}) ↔ 𝑋 = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))))
42	32, 41	mpbid 235	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → 𝑋 = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
43		cutsval 27954	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
44	34, 43	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
45	42, 44	eqtr4d 2801	. . . . 5 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
46	45	ex 417	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋) → 𝑋 = (𝐴 |s 𝐵)))
47	46	necon3d 2979	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (𝑋 ≠ (𝐴 |s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
48	47	3impia 1135	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋))
49		bdayon 27926	. . 3 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
50		bdayon 27926	. . 3 ⊢ ( bday ‘𝑋) ∈ On
51		onelpss 6403	. . 3 ⊢ ((( bday ‘(𝐴 |s 𝐵)) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋))))
52	49, 50, 51	mp2an 704	. 2 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
53	23, 48, 52	sylanbrc 594	1 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃!wreu 3367  {crab 3416   ⊆ wss 3906  ∅c0 4287  {csn 4590  ∩ cint 4913   class class class wbr 5110   “ cima 5666  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  ℩crio 7368  (class class class)co 7412   No csur 27785   bday cbday 27787   <<s cslts 27931   |s ccuts 27933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8454  df-2o 8455  df-no 27788  df-lts 27789  df-bday 27790  df-slts 27932  df-cuts 27934

This theorem is referenced by:  lesrec  27973