Theorem slelss 27860

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 27859	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ( L ‘𝐴) ⊊ ( L ‘𝐵)))
2		fveq2 6828	. . . 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 27858	. . . . . . . . . 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 4076	. . . . . . . 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 4325	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∖ ( L ‘𝐴)) = ∅
9	8	uneq1i 4113	. . . . . . . . . 10 ⊢ ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
10		0un 4345	. . . . . . . . . 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 4158	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∩ ( R ‘𝐴)) = (( R ‘𝐴) ∩ ( L ‘𝐴))
13		lltropt 27823	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
14		ssltdisj 27770	. . . . . . . . . . . 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 4429	. . . . . . . . . 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 4240	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
21		difid 4325	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∖ ( L ‘𝐵)) = ∅
22	21	uneq1i 4113	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
23		0un 4345	. . . . . . . . . 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 4158	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∩ ( R ‘𝐵)) = (( R ‘𝐵) ∩ ( L ‘𝐵))
26		lltropt 27823	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
27		ssltdisj 27770	. . . . . . . . . . . 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 4429	. . . . . . . . . 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 7370	. . . . . 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 27855	. . . . . . 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 27855	. . . . . . 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 27699	. . 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 4051	. . 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 4282   class class class wbr 5093  ‘cfv 6487  (class class class)co 7352   No csur 27584   <s cslt 27585   bday cbday 27586   ≤s csle 27689   <<s csslt 27726   |s cscut 27728   L cleft 27792   R cright 27793

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 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-1o 8391  df-2o 8392  df-no 27587  df-slt 27588  df-bday 27589  df-sle 27690  df-sslt 27727  df-scut 27729  df-made 27794  df-old 27795  df-left 27797  df-right 27798

This theorem is referenced by:  sltonold  28204  onnolt  28209