Theorem ltslpss 28138

Description: If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)

Assertion

Ref	Expression
ltslpss	⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))

Proof of Theorem ltslpss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oldno 28074	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝑋)) → 𝑥 ∈ No )
2	1	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 ∈ No )
3		simp1l1 1285	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 ∈ No )
4		simp1l2 1286	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑌 ∈ No )
5		simp3 1156	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑋)
6		simp1r 1217	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 <s 𝑌)
7	2, 3, 4, 5, 6	ltstrd 27964	. . . . . . . . 9 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑌)
8	7	3exp 1137	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday ‘𝑋)) → (𝑥 <s 𝑋 → 𝑥 <s 𝑌)))
9	8	imdistand 581	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑌)))
10		fveq2 6888	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) = ( bday ‘𝑌) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
11	10	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
12	11	adantr 486	. . . . . . . . 9 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
13	12	eleq2d 2852	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday ‘𝑋)) ↔ 𝑥 ∈ ( O ‘( bday ‘𝑌))))
14	13	anbi1d 643	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑌) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)))
15	9, 14	sylibd 242	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)))
16		leftval 28079	. . . . . . . . 9 ⊢ ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}
17	16	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋})
18	17	eleq2d 2852	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}))
19		rabid 3440	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋))
20	18, 19	bitrdi 290	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
21		leftval 28079	. . . . . . . . 9 ⊢ ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌}
22	21	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌})
23	22	eleq2d 2852	. . . . . . 7 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌}))
24		rabid 3440	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌))
25	23, 24	bitrdi 290	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑌) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)))
26	15, 20, 25	3imtr4d 297	. . . . 5 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) → 𝑥 ∈ ( L ‘𝑌)))
27	26	ssrdv 3946	. . . 4 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊆ ( L ‘𝑌))
28		ltsirr 27947	. . . . . . . . 9 ⊢ (𝑌 ∈ No → ¬ 𝑌 <s 𝑌)
29	28	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ¬ 𝑌 <s 𝑌)
30		breq1 5117	. . . . . . . . 9 ⊢ (𝑋 = 𝑌 → (𝑋 <s 𝑌 ↔ 𝑌 <s 𝑌))
31	30	notbid 321	. . . . . . . 8 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 <s 𝑌 ↔ ¬ 𝑌 <s 𝑌))
32	29, 31	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 = 𝑌 → ¬ 𝑋 <s 𝑌))
33	32	con2d 135	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 → ¬ 𝑋 = 𝑌))
34	33	imp 412	. . . . 5 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ¬ 𝑋 = 𝑌)
35		simpr 490	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( L ‘𝑋) = ( L ‘𝑌))
36		lruneq 28137	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
37	36	adantr 486	. . . . . . . . . 10 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
38	37	adantr 486	. . . . . . . . 9 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
39	38, 35	difeq12d 4085	. . . . . . . 8 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)))
40		difundir 4247	. . . . . . . . . 10 ⊢ ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
41		difid 4335	. . . . . . . . . . 11 ⊢ (( L ‘𝑋) ∖ ( L ‘𝑋)) = ∅
42	41	uneq1i 4121	. . . . . . . . . 10 ⊢ ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
43		0un 4356	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (( R ‘𝑋) ∖ ( L ‘𝑋))
44	40, 42, 43	3eqtri 2793	. . . . . . . . 9 ⊢ ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = (( R ‘𝑋) ∖ ( L ‘𝑋))
45		incom 4165	. . . . . . . . . . 11 ⊢ (( L ‘𝑋) ∩ ( R ‘𝑋)) = (( R ‘𝑋) ∩ ( L ‘𝑋))
46		lltr 28092	. . . . . . . . . . . 12 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
47		sltsdisj 28033	. . . . . . . . . . . 12 ⊢ (( L ‘𝑋) <<s ( R ‘𝑋) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
48	46, 47	mp1i 14	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
49	45, 48	eqtr3id 2815	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅)
50		disjdif2 4446	. . . . . . . . . 10 ⊢ ((( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅ → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
51	49, 50	syl 18	. . . . . . . . 9 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
52	44, 51	eqtrid 2813	. . . . . . . 8 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
53		difundir 4247	. . . . . . . . . 10 ⊢ ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
54		difid 4335	. . . . . . . . . . 11 ⊢ (( L ‘𝑌) ∖ ( L ‘𝑌)) = ∅
55	54	uneq1i 4121	. . . . . . . . . 10 ⊢ ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
56		0un 4356	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (( R ‘𝑌) ∖ ( L ‘𝑌))
57	53, 55, 56	3eqtri 2793	. . . . . . . . 9 ⊢ ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = (( R ‘𝑌) ∖ ( L ‘𝑌))
58		incom 4165	. . . . . . . . . . 11 ⊢ (( L ‘𝑌) ∩ ( R ‘𝑌)) = (( R ‘𝑌) ∩ ( L ‘𝑌))
59		lltr 28092	. . . . . . . . . . . 12 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌)
60		sltsdisj 28033	. . . . . . . . . . . 12 ⊢ (( L ‘𝑌) <<s ( R ‘𝑌) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
61	59, 60	mp1i 14	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
62	58, 61	eqtr3id 2815	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅)
63		disjdif2 4446	. . . . . . . . . 10 ⊢ ((( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅ → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
64	62, 63	syl 18	. . . . . . . . 9 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
65	57, 64	eqtrid 2813	. . . . . . . 8 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
66	39, 52, 65	3eqtr3d 2809	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( R ‘𝑋) = ( R ‘𝑌))
67	35, 66	oveq12d 7441	. . . . . 6 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑌) |s ( R ‘𝑌)))
68		simpll1 1231	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 ∈ No )
69		lrcut 28134	. . . . . . 7 ⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
70	68, 69	syl 18	. . . . . 6 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
71		simpll2 1232	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑌 ∈ No )
72		lrcut 28134	. . . . . . 7 ⊢ (𝑌 ∈ No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
73	71, 72	syl 18	. . . . . 6 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
74	67, 70, 73	3eqtr3d 2809	. . . . 5 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 = 𝑌)
75	34, 74	mtand 828	. . . 4 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ¬ ( L ‘𝑋) = ( L ‘𝑌))
76		dfpss2 4045	. . . 4 ⊢ (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑋) = ( L ‘𝑌)))
77	27, 75, 76	sylanbrc 595	. . 3 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊊ ( L ‘𝑌))
78	77	ex 418	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 → ( L ‘𝑋) ⊊ ( L ‘𝑌)))
79		dfpss3 4046	. . 3 ⊢ (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)))
80		ssdif0 4324	. . . . . . 7 ⊢ (( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) = ∅)
81	80	necon3bbii 3008	. . . . . 6 ⊢ (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅)
82		n0 4310	. . . . . 6 ⊢ ((( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
83	81, 82	bitri 278	. . . . 5 ⊢ (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
84		eldif 3918	. . . . . . 7 ⊢ (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) ↔ (𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)))
85	21	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ No → ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌})
86	85	eleq2d 2852	. . . . . . . . . . 11 ⊢ (𝑌 ∈ No → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌}))
87	86, 24	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑌 ∈ No → (𝑥 ∈ ( L ‘𝑌) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)))
88	16	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ No → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋})
89	88	eleq2d 2852	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}))
90	89, 19	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ No → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
91	90	notbid 321	. . . . . . . . . . 11 ⊢ (𝑋 ∈ No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ ¬ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
92		ianor 997	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋))
93	91, 92	bitrdi 290	. . . . . . . . . 10 ⊢ (𝑋 ∈ No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋)))
94	87, 93	bi2anan9r 651	. . . . . . . . 9 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) ↔ ((𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋))))
95	94	3adant3 1150	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) ↔ ((𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋))))
96		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday ‘𝑌)))
97		simpl3 1212	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → ( bday ‘𝑋) = ( bday ‘𝑌))
98	97	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
99	96, 98	eleqtrrd 2869	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday ‘𝑋)))
100	99	pm2.24d 152	. . . . . . . . . 10 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) → 𝑋 <s 𝑌))
101		simpll1 1231	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ∈ No )
102	96	oldnod 28077	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ No )
103	102	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 ∈ No )
104		simpll2 1232	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑌 ∈ No )
105		simpl1 1210	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑋 ∈ No )
106		lenlts 27953	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ No ∧ 𝑥 ∈ No ) → (𝑋 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑋))
107	105, 102, 106	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → (𝑋 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑋))
108	107	biimpar 483	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ≤s 𝑥)
109		simplrr 790	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 <s 𝑌)
110	101, 103, 104, 108, 109	leltstrd 27966	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 <s 𝑌)
111	110	ex 418	. . . . . . . . . 10 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → (¬ 𝑥 <s 𝑋 → 𝑋 <s 𝑌))
112	100, 111	jaod 873	. . . . . . . . 9 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → ((¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋) → 𝑋 <s 𝑌))
113	112	expimpd 459	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (((𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋)) → 𝑋 <s 𝑌))
114	95, 113	sylbid 243	. . . . . . 7 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 <s 𝑌))
115	84, 114	biimtrid 245	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
116	115	exlimdv 1966	. . . . 5 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
117	83, 116	biimtrid 245	. . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) → 𝑋 <s 𝑌))
118	117	adantld 496	. . 3 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ((( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)) → 𝑋 <s 𝑌))
119	79, 118	biimtrid 245	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ⊊ ( L ‘𝑌) → 𝑋 <s 𝑌))
120	78, 119	impbid 215	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2961  {crab 3419   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908   ⊊ wpss 3909  ∅c0 4289   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423   No csur 27841   <s clts 27842   bday cbday 27843   ≤s cles 27945   <<s cslts 27987   |s ccuts 27989   O cold 28053   L cleft 28055   R cright 28056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-made 28057  df-old 28058  df-left 28060  df-right 28061

This theorem is used by:  leslss  28139