Theorem leslss 28182

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 28181	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ( L ‘𝐴) ⊊ ( L ‘𝐵)))
2		fveq2 6882	. . . 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 28180	. . . . . . . . . 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 4078	. . . . . . . 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 4240	. . . . . . . . . 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 4114	. . . . . . . . . 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 2789	. . . . . . . . 9 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = (( R ‘𝐴) ∖ ( L ‘𝐴))
12		incom 4158	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∩ ( R ‘𝐴)) = (( R ‘𝐴) ∩ ( L ‘𝐴))
13		lltr 28135	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
14		sltsdisj 28076	. . . . . . . . . . . 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 2811	. . . . . . . . . 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 2809	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
20		difundir 4240	. . . . . . . . . 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 4114	. . . . . . . . . 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 2789	. . . . . . . . 9 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = (( R ‘𝐵) ∖ ( L ‘𝐵))
25		incom 4158	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∩ ( R ‘𝐵)) = (( R ‘𝐵) ∩ ( L ‘𝐵))
26		lltr 28135	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
27		sltsdisj 28076	. . . . . . . . . . . 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 2811	. . . . . . . . . 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 2809	. . . . . . . 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 2805	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ( R ‘𝐴) = ( R ‘𝐵))
34	3, 33	oveq12d 7435	. . . . . 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 28177	. . . . . . 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 28177	. . . . . . 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 2805	. . . . 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 27998	. . 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 4053	. . 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 2145   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417   No csur 27884   <s clts 27885   bday cbday 27886   ≤s cles 27988   <<s cslts 28030   |s ccuts 28032   L cleft 28098   R cright 28099

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033  df-made 28100  df-old 28101  df-left 28103  df-right 28104

This theorem is used by:  ltonold  28534  onnolt  28539