Theorem leslss 28328

Description: If two surreals 𝐴 and 𝐵 share a birthday, then 𝐴 ≤s 𝐵 if and only if the left set of 𝐴 is a non-strict subset of the left set of 𝐵. (Contributed by Scott Fenton, 21-Mar-2025.)

Assertion

Ref	Expression
leslss	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 ≤s 𝐵 ↔ (L‘𝐴) ⊆ (L‘𝐵)))

Proof of Theorem leslss

Step	Hyp	Ref	Expression
1		ltslpss 28327	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 <s 𝐵 ↔ (L‘𝐴) ⊊ (L‘𝐵)))
2		fveq2 6911	. . . 4 ⊢ (𝐴 = 𝐵 → (L‘𝐴) = (L‘𝐵))
3		simpr 490	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (L‘𝐴) = (L‘𝐵))
4		lruneq 28326	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → ((L‘𝐴) ∪ (R‘𝐴)) = ((L‘𝐵) ∪ (R‘𝐵)))
5	4	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐴) ∪ (R‘𝐴)) = ((L‘𝐵) ∪ (R‘𝐵)))
6	5, 3	difeq12d 4067	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (((L‘𝐴) ∪ (R‘𝐴)) ∖ (L‘𝐴)) = (((L‘𝐵) ∪ (R‘𝐵)) ∖ (L‘𝐵)))
7		difundir 4239	. . . . . . . . . 10 ⊢ (((L‘𝐴) ∪ (R‘𝐴)) ∖ (L‘𝐴)) = (((L‘𝐴) ∖ (L‘𝐴)) ∪ ((R‘𝐴) ∖ (L‘𝐴)))
8		difid 4328	. . . . . . . . . . 11 ⊢ ((L‘𝐴) ∖ (L‘𝐴)) = ∅
9	8	uneq1i 4166	. . . . . . . . . 10 ⊢ (((L‘𝐴) ∖ (L‘𝐴)) ∪ ((R‘𝐴) ∖ (L‘𝐴))) = (∅ ∪ ((R‘𝐴) ∖ (L‘𝐴)))
10		0un 4349	. . . . . . . . . 10 ⊢ (∅ ∪ ((R‘𝐴) ∖ (L‘𝐴))) = ((R‘𝐴) ∖ (L‘𝐴))
11	7, 9, 10	3eqtri 2790	. . . . . . . . 9 ⊢ (((L‘𝐴) ∪ (R‘𝐴)) ∖ (L‘𝐴)) = ((R‘𝐴) ∖ (L‘𝐴))
12		incom 4109	. . . . . . . . . . 11 ⊢ ((L‘𝐴) ∩ (R‘𝐴)) = ((R‘𝐴) ∩ (L‘𝐴))
13		lltr 28281	. . . . . . . . . . . 12 ⊢ (L‘𝐴) <<s (R‘𝐴)
14		sltsdisj 28222	. . . . . . . . . . . 12 ⊢ ((L‘𝐴) <<s (R‘𝐴) → ((L‘𝐴) ∩ (R‘𝐴)) = ∅)
15	13, 14	mp1i 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐴) ∩ (R‘𝐴)) = ∅)
16	12, 15	eqtr3id 2812	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐴) ∩ (L‘𝐴)) = ∅)
17		disjdif2 4439	. . . . . . . . . 10 ⊢ (((R‘𝐴) ∩ (L‘𝐴)) = ∅ → ((R‘𝐴) ∖ (L‘𝐴)) = (R‘𝐴))
18	16, 17	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐴) ∖ (L‘𝐴)) = (R‘𝐴))
19	11, 18	eqtrid 2810	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (((L‘𝐴) ∪ (R‘𝐴)) ∖ (L‘𝐴)) = (R‘𝐴))
20		difundir 4239	. . . . . . . . . 10 ⊢ (((L‘𝐵) ∪ (R‘𝐵)) ∖ (L‘𝐵)) = (((L‘𝐵) ∖ (L‘𝐵)) ∪ ((R‘𝐵) ∖ (L‘𝐵)))
21		difid 4328	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∖ (L‘𝐵)) = ∅
22	21	uneq1i 4166	. . . . . . . . . 10 ⊢ (((L‘𝐵) ∖ (L‘𝐵)) ∪ ((R‘𝐵) ∖ (L‘𝐵))) = (∅ ∪ ((R‘𝐵) ∖ (L‘𝐵)))
23		0un 4349	. . . . . . . . . 10 ⊢ (∅ ∪ ((R‘𝐵) ∖ (L‘𝐵))) = ((R‘𝐵) ∖ (L‘𝐵))
24	20, 22, 23	3eqtri 2790	. . . . . . . . 9 ⊢ (((L‘𝐵) ∪ (R‘𝐵)) ∖ (L‘𝐵)) = ((R‘𝐵) ∖ (L‘𝐵))
25		incom 4109	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∩ (R‘𝐵)) = ((R‘𝐵) ∩ (L‘𝐵))
26		lltr 28281	. . . . . . . . . . . 12 ⊢ (L‘𝐵) <<s (R‘𝐵)
27		sltsdisj 28222	. . . . . . . . . . . 12 ⊢ ((L‘𝐵) <<s (R‘𝐵) → ((L‘𝐵) ∩ (R‘𝐵)) = ∅)
28	26, 27	mp1i 14	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐵) ∩ (R‘𝐵)) = ∅)
29	25, 28	eqtr3id 2812	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐵) ∩ (L‘𝐵)) = ∅)
30		disjdif2 4439	. . . . . . . . . 10 ⊢ (((R‘𝐵) ∩ (L‘𝐵)) = ∅ → ((R‘𝐵) ∖ (L‘𝐵)) = (R‘𝐵))
31	29, 30	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐵) ∖ (L‘𝐵)) = (R‘𝐵))
32	24, 31	eqtrid 2810	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (((L‘𝐵) ∪ (R‘𝐵)) ∖ (L‘𝐵)) = (R‘𝐵))
33	6, 19, 32	3eqtr3d 2806	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (R‘𝐴) = (R‘𝐵))
34	3, 33	oveq12d 7464	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐴) |s (R‘𝐴)) = ((L‘𝐵) |s (R‘𝐵)))
35		simpl1 1210	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → 𝐴 ∈ No)
36		lrcut 28323	. . . . . . 7 ⊢ (𝐴 ∈ No → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
37	35, 36	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐴) |s (R‘𝐴)) = 𝐴)
38		simpl2 1211	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → 𝐵 ∈ No)
39		lrcut 28323	. . . . . . 7 ⊢ (𝐵 ∈ No → ((L‘𝐵) |s (R‘𝐵)) = 𝐵)
40	38, 39	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((L‘𝐵) |s (R‘𝐵)) = 𝐵)
41	34, 37, 40	3eqtr3d 2806	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → 𝐴 = 𝐵)
42	41	ex 418	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → ((L‘𝐴) = (L‘𝐵) → 𝐴 = 𝐵))
43	2, 42	impbid2 229	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 = 𝐵 ↔ (L‘𝐴) = (L‘𝐵)))
44	1, 43	orbi12d 932	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → ((𝐴 <s 𝐵 ∨ 𝐴 = 𝐵) ↔ ((L‘𝐴) ⊊ (L‘𝐵) ∨ (L‘𝐴) = (L‘𝐵))))
45		lesloe 28144	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
46	45	3adant3 1150	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
47		sspss 4032	. . 3 ⊢ ((L‘𝐴) ⊆ (L‘𝐵) ↔ ((L‘𝐴) ⊊ (L‘𝐵) ∨ (L‘𝐴) = (L‘𝐵)))
48	47	a1i 11	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → ((L‘𝐴) ⊆ (L‘𝐵) ↔ ((L‘𝐴) ⊊ (L‘𝐵) ∨ (L‘𝐴) = (L‘𝐵))))
49	44, 46, 48	3bitr4d 314	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 ≤s 𝐵 ↔ (L‘𝐴) ⊆ (L‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ⊊ wpss 4016   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   <s clts 28031  bdaycbday 28032   ≤s cles 28134   <<s cslts 28176   |s ccuts 28178  Lcleft 28244  Rcright 28245

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-made 28246  df-old 28247  df-left 28249  df-right 28250

This theorem is used by:  ltonold  28680  onnolt  28685