Theorem ltslpss 27900

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 27836	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ( O ‘( bday ‘𝑋)) → 𝑥 ∈ No )
2	1	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 ∈ No )
3		simp1l1 1268	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 ∈ No )
4		simp1l2 1269	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑌 ∈ No )
5		simp3 1139	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑋)
6		simp1r 1200	. . . . . . . . . 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 27727	. . . . . . . . 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 1120	. . . . . . . 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 570	. . . . . . 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 6840	. . . . . . . . . . 11 ⊢ (( bday ‘𝑋) = ( bday ‘𝑌) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
11	10	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
12	11	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( O ‘( bday ‘𝑋)) = ( O ‘( bday ‘𝑌)))
13	12	eleq2d 2822	. . . . . . . 8 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday ‘𝑋)) ↔ 𝑥 ∈ ( O ‘( bday ‘𝑌))))
14	13	anbi1d 632	. . . . . . 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 239	. . . . . 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 27841	. . . . . . . . 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 2822	. . . . . . 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 3410	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋))
20	18, 19	bitrdi 287	. . . . . 6 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
21		leftval 27841	. . . . . . . . 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 2822	. . . . . . 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 3410	. . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌} ↔ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌))
25	23, 24	bitrdi 287	. . . . . 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 294	. . . . 5 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) → 𝑥 ∈ ( L ‘𝑌)))
27	26	ssrdv 3927	. . . 4 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊆ ( L ‘𝑌))
28		ltsirr 27710	. . . . . . . . 9 ⊢ (𝑌 ∈ No → ¬ 𝑌 <s 𝑌)
29	28	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ¬ 𝑌 <s 𝑌)
30		breq1 5088	. . . . . . . . 9 ⊢ (𝑋 = 𝑌 → (𝑋 <s 𝑌 ↔ 𝑌 <s 𝑌))
31	30	notbid 318	. . . . . . . 8 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 <s 𝑌 ↔ ¬ 𝑌 <s 𝑌))
32	29, 31	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 = 𝑌 → ¬ 𝑋 <s 𝑌))
33	32	con2d 134	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 → ¬ 𝑋 = 𝑌))
34	33	imp 406	. . . . 5 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ¬ 𝑋 = 𝑌)
35		simpr 484	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( L ‘𝑋) = ( L ‘𝑌))
36		lruneq 27899	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
37	36	adantr 480	. . . . . . . . . 10 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
38	37	adantr 480	. . . . . . . . 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 4067	. . . . . . . 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 4231	. . . . . . . . . 10 ⊢ ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
41		difid 4316	. . . . . . . . . . 11 ⊢ (( L ‘𝑋) ∖ ( L ‘𝑋)) = ∅
42	41	uneq1i 4104	. . . . . . . . . 10 ⊢ ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
43		0un 4336	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (( R ‘𝑋) ∖ ( L ‘𝑋))
44	40, 42, 43	3eqtri 2763	. . . . . . . . 9 ⊢ ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = (( R ‘𝑋) ∖ ( L ‘𝑋))
45		incom 4149	. . . . . . . . . . 11 ⊢ (( L ‘𝑋) ∩ ( R ‘𝑋)) = (( R ‘𝑋) ∩ ( L ‘𝑋))
46		lltr 27854	. . . . . . . . . . . 12 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋)
47		sltsdisj 27795	. . . . . . . . . . . 12 ⊢ (( L ‘𝑋) <<s ( R ‘𝑋) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
48	46, 47	mp1i 13	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
49	45, 48	eqtr3id 2785	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅)
50		disjdif2 4420	. . . . . . . . . 10 ⊢ ((( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅ → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
51	49, 50	syl 17	. . . . . . . . 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 2783	. . . . . . . 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 4231	. . . . . . . . . 10 ⊢ ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
54		difid 4316	. . . . . . . . . . 11 ⊢ (( L ‘𝑌) ∖ ( L ‘𝑌)) = ∅
55	54	uneq1i 4104	. . . . . . . . . 10 ⊢ ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
56		0un 4336	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (( R ‘𝑌) ∖ ( L ‘𝑌))
57	53, 55, 56	3eqtri 2763	. . . . . . . . 9 ⊢ ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = (( R ‘𝑌) ∖ ( L ‘𝑌))
58		incom 4149	. . . . . . . . . . 11 ⊢ (( L ‘𝑌) ∩ ( R ‘𝑌)) = (( R ‘𝑌) ∩ ( L ‘𝑌))
59		lltr 27854	. . . . . . . . . . . 12 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌)
60		sltsdisj 27795	. . . . . . . . . . . 12 ⊢ (( L ‘𝑌) <<s ( R ‘𝑌) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
61	59, 60	mp1i 13	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
62	58, 61	eqtr3id 2785	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅)
63		disjdif2 4420	. . . . . . . . . 10 ⊢ ((( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅ → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
64	62, 63	syl 17	. . . . . . . . 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 2783	. . . . . . . 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 2779	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( R ‘𝑋) = ( R ‘𝑌))
67	35, 66	oveq12d 7385	. . . . . 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 1214	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 ∈ No )
69		lrcut 27896	. . . . . . 7 ⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
70	68, 69	syl 17	. . . . . 6 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
71		simpll2 1215	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑌 ∈ No )
72		lrcut 27896	. . . . . . 7 ⊢ (𝑌 ∈ No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
73	71, 72	syl 17	. . . . . 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 2779	. . . . 5 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 = 𝑌)
75	34, 74	mtand 816	. . . 4 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ¬ ( L ‘𝑋) = ( L ‘𝑌))
76		dfpss2 4028	. . . 4 ⊢ (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑋) = ( L ‘𝑌)))
77	27, 75, 76	sylanbrc 584	. . 3 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊊ ( L ‘𝑌))
78	77	ex 412	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 → ( L ‘𝑋) ⊊ ( L ‘𝑌)))
79		dfpss3 4029	. . 3 ⊢ (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)))
80		ssdif0 4306	. . . . . . 7 ⊢ (( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) = ∅)
81	80	necon3bbii 2979	. . . . . 6 ⊢ (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅)
82		n0 4293	. . . . . 6 ⊢ ((( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
83	81, 82	bitri 275	. . . . 5 ⊢ (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
84		eldif 3899	. . . . . . 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 2822	. . . . . . . . . . 11 ⊢ (𝑌 ∈ No → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑥 <s 𝑌}))
87	86, 24	bitrdi 287	. . . . . . . . . 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 2822	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ No → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday ‘𝑋)) ∣ 𝑥 <s 𝑋}))
90	89, 19	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ No → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
91	90	notbid 318	. . . . . . . . . . 11 ⊢ (𝑋 ∈ No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ ¬ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋)))
92		ianor 984	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ ( O ‘( bday ‘𝑋)) ∧ 𝑥 <s 𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋))
93	91, 92	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑋 ∈ No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday ‘𝑋)) ∨ ¬ 𝑥 <s 𝑋)))
94	87, 93	bi2anan9r 640	. . . . . . . . 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 1133	. . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday ‘𝑌)))
97		simpl3 1195	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → ( bday ‘𝑋) = ( bday ‘𝑌))
98	97	fveq2d 6844	. . . . . . . . . . . 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 2839	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday ‘𝑋)))
100	99	pm2.24d 151	. . . . . . . . . 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 1214	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ∈ No )
102	96	oldnod 27839	. . . . . . . . . . . . 13 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ No )
103	102	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 ∈ No )
104		simpll2 1215	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑌 ∈ No )
105		simpl1 1193	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑋 ∈ No )
106		lenlts 27716	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ No ∧ 𝑥 ∈ No ) → (𝑋 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑋))
107	105, 102, 106	syl2anc 585	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) ∧ (𝑥 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ≤s 𝑥)
109		simplrr 778	. . . . . . . . . . . 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 27729	. . . . . . . . . . 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 412	. . . . . . . . . 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 860	. . . . . . . . 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 453	. . . . . . . 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 240	. . . . . . 7 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 <s 𝑌))
115	84, 114	biimtrid 242	. . . . . 6 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
116	115	exlimdv 1935	. . . . 5 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
117	83, 116	biimtrid 242	. . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) → 𝑋 <s 𝑌))
118	117	adantld 490	. . 3 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → ((( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)) → 𝑋 <s 𝑌))
119	79, 118	biimtrid 242	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ⊊ ( L ‘𝑌) → 𝑋 <s 𝑌))
120	78, 119	impbid 212	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  {crab 3389   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889   ⊊ wpss 3890  ∅c0 4273   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   bday cbday 27605   ≤s cles 27708   <<s cslts 27749   |s ccuts 27751   O cold 27815   L cleft 27817   R cright 27818

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823

This theorem is referenced by:  leslss  27901