Theorem ltslpss 28299

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 28235	. . . . . . . . . . 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 28125	. . . . . . . . 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 6885	. . . . . . . . . . 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 2846	. . . . . . . 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 28240	. . . . . . . . 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 2846	. . . . . . 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 3432	. . . . . . 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 28240	. . . . . . . . 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 2846	. . . . . . 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 3432	. . . . . . 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 3914	. . . 4 ⊢ (((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ (bday‘𝑋) = (bday‘𝑌)) ∧ 𝑋 <s 𝑌) → (L‘𝑋) ⊆ (L‘𝑌))
28		ltsirr 28108	. . . . . . . . 9 ⊢ (𝑌 ∈ No → ¬ 𝑌 <s 𝑌)
29	28	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ (bday‘𝑋) = (bday‘𝑌)) → ¬ 𝑌 <s 𝑌)
30		breq1 5105	. . . . . . . . 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 28298	. . . . . . . . . . 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 4064	. . . . . . . 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 4236	. . . . . . . . . 10 ⊢ (((L‘𝑋) ∪ (R‘𝑋)) ∖ (L‘𝑋)) = (((L‘𝑋) ∖ (L‘𝑋)) ∪ ((R‘𝑋) ∖ (L‘𝑋)))
41		difid 4324	. . . . . . . . . . 11 ⊢ ((L‘𝑋) ∖ (L‘𝑋)) = ∅
42	41	uneq1i 4163	. . . . . . . . . 10 ⊢ (((L‘𝑋) ∖ (L‘𝑋)) ∪ ((R‘𝑋) ∖ (L‘𝑋))) = (∅ ∪ ((R‘𝑋) ∖ (L‘𝑋)))
43		0un 4345	. . . . . . . . . 10 ⊢ (∅ ∪ ((R‘𝑋) ∖ (L‘𝑋))) = ((R‘𝑋) ∖ (L‘𝑋))
44	40, 42, 43	3eqtri 2787	. . . . . . . . 9 ⊢ (((L‘𝑋) ∪ (R‘𝑋)) ∖ (L‘𝑋)) = ((R‘𝑋) ∖ (L‘𝑋))
45		incom 4106	. . . . . . . . . . 11 ⊢ ((L‘𝑋) ∩ (R‘𝑋)) = ((R‘𝑋) ∩ (L‘𝑋))
46		lltr 28253	. . . . . . . . . . . 12 ⊢ (L‘𝑋) <<s (R‘𝑋)
47		sltsdisj 28194	. . . . . . . . . . . 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 2809	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ (bday‘𝑋) = (bday‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ (L‘𝑋) = (L‘𝑌)) → ((R‘𝑋) ∩ (L‘𝑋)) = ∅)
50		disjdif2 4435	. . . . . . . . . 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 2807	. . . . . . . 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 4236	. . . . . . . . . 10 ⊢ (((L‘𝑌) ∪ (R‘𝑌)) ∖ (L‘𝑌)) = (((L‘𝑌) ∖ (L‘𝑌)) ∪ ((R‘𝑌) ∖ (L‘𝑌)))
54		difid 4324	. . . . . . . . . . 11 ⊢ ((L‘𝑌) ∖ (L‘𝑌)) = ∅
55	54	uneq1i 4163	. . . . . . . . . 10 ⊢ (((L‘𝑌) ∖ (L‘𝑌)) ∪ ((R‘𝑌) ∖ (L‘𝑌))) = (∅ ∪ ((R‘𝑌) ∖ (L‘𝑌)))
56		0un 4345	. . . . . . . . . 10 ⊢ (∅ ∪ ((R‘𝑌) ∖ (L‘𝑌))) = ((R‘𝑌) ∖ (L‘𝑌))
57	53, 55, 56	3eqtri 2787	. . . . . . . . 9 ⊢ (((L‘𝑌) ∪ (R‘𝑌)) ∖ (L‘𝑌)) = ((R‘𝑌) ∖ (L‘𝑌))
58		incom 4106	. . . . . . . . . . 11 ⊢ ((L‘𝑌) ∩ (R‘𝑌)) = ((R‘𝑌) ∩ (L‘𝑌))
59		lltr 28253	. . . . . . . . . . . 12 ⊢ (L‘𝑌) <<s (R‘𝑌)
60		sltsdisj 28194	. . . . . . . . . . . 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 2809	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ (bday‘𝑋) = (bday‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ (L‘𝑋) = (L‘𝑌)) → ((R‘𝑌) ∩ (L‘𝑌)) = ∅)
63		disjdif2 4435	. . . . . . . . . 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 2807	. . . . . . . 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 2803	. . . . . . 7 ⊢ ((((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ (bday‘𝑋) = (bday‘𝑌)) ∧ 𝑋 <s 𝑌) ∧ (L‘𝑋) = (L‘𝑌)) → (R‘𝑋) = (R‘𝑌))
67	35, 66	oveq12d 7438	. . . . . 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 28295	. . . . . . 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 28295	. . . . . . 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 2803	. . . . 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 4015	. . . 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 4016	. . 3 ⊢ ((L‘𝑋) ⊊ (L‘𝑌) ↔ ((L‘𝑋) ⊆ (L‘𝑌) ∧ ¬ (L‘𝑌) ⊆ (L‘𝑋)))
80		ssdif0 4313	. . . . . . 7 ⊢ ((L‘𝑌) ⊆ (L‘𝑋) ↔ ((L‘𝑌) ∖ (L‘𝑋)) = ∅)
81	80	necon3bbii 3002	. . . . . 6 ⊢ (¬ (L‘𝑌) ⊆ (L‘𝑋) ↔ ((L‘𝑌) ∖ (L‘𝑋)) ≠ ∅)
82		n0 4299	. . . . . 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 4049	. . . . . . 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 2846	. . . . . . . . . . 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 2846	. . . . . . . . . . . . 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 6889	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 28238	. . . . . . . . . . . . 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 28114	. . . . . . . . . . . . . 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 28127	. . . . . . . . . . 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 1965	. . . . 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 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  {crab 3412   ⊆ wss 3895   ⊊ wpss 4013   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <s clts 28003  bdaycbday 28004   ≤s cles 28106   <<s cslts 28148   |s ccuts 28150  Ocold 28214  Lcleft 28216  Rcright 28217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-left 28221  df-right 28222

This theorem is used by:  leslss  28300