Theorem scutbdaylt 27881

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

Proof of Theorem scutbdaylt

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

Step	Hyp	Ref	Expression
1		simp2l 1199	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s {𝑋})
2		simp2r 1200	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} <<s 𝐵)
3		snnzg 4799	. . . . . 6 ⊢ (𝑋 ∈ No → {𝑋} ≠ ∅)
4	3	3ad2ant1 1133	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → {𝑋} ≠ ∅)
5		sslttr 27870	. . . . 5 ⊢ ((𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵 ∧ {𝑋} ≠ ∅) → 𝐴 <<s 𝐵)
6	1, 2, 4, 5	syl3anc 1371	. . . 4 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝐴 <<s 𝐵)
7		scutbday 27867	. . . 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 27836	. . . . 5 ⊢ bday Fn No
10		ssrab2 4103	. . . . 5 ⊢ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ⊆ No
11		simp1 1136	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ No )
12		simp2 1137	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵))
13		sneq 4658	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → {𝑦} = {𝑋})
14	13	breq2d 5178	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {𝑋}))
15	13	breq1d 5176	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ({𝑦} <<s 𝐵 ↔ {𝑋} <<s 𝐵))
16	14, 15	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
17	16	elrab 3708	. . . . . 6 ⊢ (𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ (𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)))
18	11, 12, 17	sylanbrc 582	. . . . 5 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → 𝑋 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
19		fnfvima 7270	. . . . 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 1465	. . . 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 4987	. . . 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 4047	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋))
24		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → 𝐴 <<s {𝑋})
25		simprr 772	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 2742	. . . . . . . . 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 2747	. . . . . . . . 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 27862	. . . . . . . . 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 6929	. . . . . . . . . 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 7430	. . . . . . . . 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 2747	. . . . . . . . 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 684	. . . . . . 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		scutval 27863	. . . . . . 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 2783	. . . . 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 2967	. . 3 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵)) → (𝑋 ≠ (𝐴 |s 𝐵) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
48	47	3impia 1117	. 2 ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋))
49		bdayelon 27839	. . 3 ⊢ ( bday ‘(𝐴 |s 𝐵)) ∈ On
50		bdayelon 27839	. . 3 ⊢ ( bday ‘𝑋) ∈ On
51		onelpss 6435	. . 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 691	. 2 ⊢ (( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋) ↔ (( bday ‘(𝐴 |s 𝐵)) ⊆ ( bday ‘𝑋) ∧ ( bday ‘(𝐴 |s 𝐵)) ≠ ( bday ‘𝑋)))
53	23, 48, 52	sylanbrc 582	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 1537   ∈ wcel 2108   ≠ wne 2946  ∃!wreu 3386  {crab 3443   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∩ cint 4970   class class class wbr 5166   “ cima 5703  Oncon0 6395   Fn wfn 6568  ‘cfv 6573  ℩crio 7403  (class class class)co 7448   No csur 27702   bday cbday 27704   <<s csslt 27843   |s cscut 27845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846

This theorem is referenced by:  slerec  27882