Theorem cutbdaylt 27806

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 1201	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s {𝑋})
2		simp2r 1202	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} <<s 𝐵)
3		snnzg 4733	. . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} ≠ ∅)
4	3	3ad2ant1 1134	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} ≠ ∅)
5		sltstr 27795	. . . . 5 ⊢ ((𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵 ∧ {𝑋} ≠ ∅) → 𝐴 <<s 𝐵)
6	1, 2, 4, 5	syl3anc 1374	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
7		cutbday 27792	. . . 4 ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
8	6, 7	syl 17	. . 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 27757	. . . . 5 ⊢ bday Fn No
10		ssrab2 4034	. . . . 5 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
11		simp1 1137	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ No )
12		simp2 1138	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵))
13		sneq 4592	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → {𝑦} = {𝑋})
14	13	breq2d 5112	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑋}))
15	13	breq1d 5110	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ({𝑦} <<s 𝐵 ↔ {𝑋} <<s 𝐵))
16	14, 15	anbi12d 633	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
17	16	elrab 3648	. . . . . 6 ⊢ (𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
18	11, 12, 17	sylanbrc 584	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
19		fnfvima 7189	. . . . 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 1468	. . . 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 4920	. . . 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 17	. . 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 3970	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋))
24		simprl 771	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s {𝑋})
25		simprr 773	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} <<s 𝐵)
26	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → {𝑋} ≠ ∅)
27	24, 25, 26, 5	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s 𝐵)
28	27, 7	syl 17	. . . . . . . . . 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 2739	. . . . . . . . 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 2744	. . . . . . . . 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 287	. . . . . . . 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 476	. . . . . . 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 228	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
34	27	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → 𝐴 <<s 𝐵)
35		conway 27787	. . . . . . . . 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 17	. . . . . . . 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 6851	. . . . . . . . . 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 7350	. . . . . . . . 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 2744	. . . . . . . . 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 287	. . . . . . . 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 686	. . . . . . 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 232	. . . . . 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 27788	. . . . . . 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 17	. . . . . 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 2775	. . . . 5 ⊢ (((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) ∧ ( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋)) → 𝑋 = (𝐴 |s 𝐵))
46	45	ex 412	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (( bday ‘(𝐴 |s 𝐵)) = ( bday ‘𝑋) → 𝑋 = (𝐴 |s 𝐵)))
47	46	necon3d 2954	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (𝑋 ≠ (𝐴 |s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
48	47	3impia 1118	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋))
49		bdayon 27760	. . 3 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
50		bdayon 27760	. . 3 ⊢ ( bday ‘𝑋) ∈ On
51		onelpss 6365	. . 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 693	. 2 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
53	23, 48, 52	sylanbrc 584	1 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃!wreu 3350  {crab 3401   ⊆ wss 3903  ∅c0 4287  {csn 4582  ∩ cint 4904   class class class wbr 5100   “ cima 5635  Oncon0 6325   Fn wfn 6495  ‘cfv 6500  ℩crio 7324  (class class class)co 7368   No csur 27619   bday cbday 27621   <<s cslts 27765   |s ccuts 27767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-slts 27766  df-cuts 27768

This theorem is referenced by:  lesrec  27807