Theorem slelss 27849

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

Proof of Theorem slelss

Step	Hyp	Ref	Expression
1		sltlpss 27848	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ( L ‘𝐴) ⊊ ( L ‘𝐵)))
2		fveq2 6817	. . . 4 ⊢ (𝐴 = 𝐵 → ( L ‘𝐴) = ( L ‘𝐵))
3		simpr 484	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ( L ‘𝐴) = ( L ‘𝐵))
4		lruneq 27847	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
5	4	adantr 480	. . . . . . . . 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 4072	. . . . . . . 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 4236	. . . . . . . . . 10 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
8		difid 4321	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∖ ( L ‘𝐴)) = ∅
9	8	uneq1i 4109	. . . . . . . . . 10 ⊢ ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
10		0un 4341	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (( R ‘𝐴) ∖ ( L ‘𝐴))
11	7, 9, 10	3eqtri 2758	. . . . . . . . 9 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = (( R ‘𝐴) ∖ ( L ‘𝐴))
12		incom 4154	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∩ ( R ‘𝐴)) = (( R ‘𝐴) ∩ ( L ‘𝐴))
13		lltropt 27812	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
14		ssltdisj 27759	. . . . . . . . . . . 12 ⊢ (( L ‘𝐴) <<s ( R ‘𝐴) → (( L ‘𝐴) ∩ ( R ‘𝐴)) = ∅)
15	13, 14	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐴) ∩ ( R ‘𝐴)) = ∅)
16	12, 15	eqtr3id 2780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐴) ∩ ( L ‘𝐴)) = ∅)
17		disjdif2 4425	. . . . . . . . . 10 ⊢ ((( R ‘𝐴) ∩ ( L ‘𝐴)) = ∅ → (( R ‘𝐴) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐴) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
19	11, 18	eqtrid 2778	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
20		difundir 4236	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
21		difid 4321	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∖ ( L ‘𝐵)) = ∅
22	21	uneq1i 4109	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
23		0un 4341	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (( R ‘𝐵) ∖ ( L ‘𝐵))
24	20, 22, 23	3eqtri 2758	. . . . . . . . 9 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = (( R ‘𝐵) ∖ ( L ‘𝐵))
25		incom 4154	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∩ ( R ‘𝐵)) = (( R ‘𝐵) ∩ ( L ‘𝐵))
26		lltropt 27812	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
27		ssltdisj 27759	. . . . . . . . . . . 12 ⊢ (( L ‘𝐵) <<s ( R ‘𝐵) → (( L ‘𝐵) ∩ ( R ‘𝐵)) = ∅)
28	26, 27	mp1i 13	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐵) ∩ ( R ‘𝐵)) = ∅)
29	25, 28	eqtr3id 2780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐵) ∩ ( L ‘𝐵)) = ∅)
30		disjdif2 4425	. . . . . . . . . 10 ⊢ ((( R ‘𝐵) ∩ ( L ‘𝐵)) = ∅ → (( R ‘𝐵) ∖ ( L ‘𝐵)) = ( R ‘𝐵))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐵) ∖ ( L ‘𝐵)) = ( R ‘𝐵))
32	24, 31	eqtrid 2778	. . . . . . . 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 2774	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ( R ‘𝐴) = ( R ‘𝐵))
34	3, 33	oveq12d 7359	. . . . . 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 1192	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐴 ∈ No )
36		lrcut 27844	. . . . . . 7 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
37	35, 36	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
38		simpl2 1193	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐵 ∈ No )
39		lrcut 27844	. . . . . . 7 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
40	38, 39	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
41	34, 37, 40	3eqtr3d 2774	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐴 = 𝐵)
42	41	ex 412	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (( L ‘𝐴) = ( L ‘𝐵) → 𝐴 = 𝐵))
43	2, 42	impbid2 226	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 = 𝐵 ↔ ( L ‘𝐴) = ( L ‘𝐵)))
44	1, 43	orbi12d 918	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → ((𝐴 <s 𝐵 ∨ 𝐴 = 𝐵) ↔ (( L ‘𝐴) ⊊ ( L ‘𝐵) ∨ ( L ‘𝐴) = ( L ‘𝐵))))
45		sleloe 27688	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
46	45	3adant3 1132	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
47		sspss 4047	. . 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 311	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897   ⊊ wpss 3898  ∅c0 4278   class class class wbr 5086  ‘cfv 6476  (class class class)co 7341   No csur 27573   <s cslt 27574   bday cbday 27575   ≤s csle 27678   <<s csslt 27715   |s cscut 27717   L cleft 27781   R cright 27782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-no 27576  df-slt 27577  df-bday 27578  df-sle 27679  df-sslt 27716  df-scut 27718  df-made 27783  df-old 27784  df-left 27786  df-right 27787

This theorem is referenced by:  sltonold  28193  onnolt  28198