Theorem leslss 28300

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 28299	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) → (𝐴 <s 𝐵 ↔ (L‘𝐴) ⊊ (L‘𝐵)))
2		fveq2 6885	. . . 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 28298	. . . . . . . . . 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 4064	. . . . . . . 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 4324	. . . . . . . . . . 11 ⊢ ((L‘𝐴) ∖ (L‘𝐴)) = ∅
9	8	uneq1i 4163	. . . . . . . . . 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 2787	. . . . . . . . 9 ⊢ (((L‘𝐴) ∪ (R‘𝐴)) ∖ (L‘𝐴)) = ((R‘𝐴) ∖ (L‘𝐴))
12		incom 4106	. . . . . . . . . . 11 ⊢ ((L‘𝐴) ∩ (R‘𝐴)) = ((R‘𝐴) ∩ (L‘𝐴))
13		lltr 28253	. . . . . . . . . . . 12 ⊢ (L‘𝐴) <<s (R‘𝐴)
14		sltsdisj 28194	. . . . . . . . . . . 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 2809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐴) ∩ (L‘𝐴)) = ∅)
17		disjdif2 4435	. . . . . . . . . 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 2807	. . . . . . . 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 4324	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∖ (L‘𝐵)) = ∅
22	21	uneq1i 4163	. . . . . . . . . 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 2787	. . . . . . . . 9 ⊢ (((L‘𝐵) ∪ (R‘𝐵)) ∖ (L‘𝐵)) = ((R‘𝐵) ∖ (L‘𝐵))
25		incom 4106	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∩ (R‘𝐵)) = ((R‘𝐵) ∩ (L‘𝐵))
26		lltr 28253	. . . . . . . . . . . 12 ⊢ (L‘𝐵) <<s (R‘𝐵)
27		sltsdisj 28194	. . . . . . . . . . . 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 2809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → ((R‘𝐵) ∩ (L‘𝐵)) = ∅)
30		disjdif2 4435	. . . . . . . . . 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 2807	. . . . . . . 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 2803	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (bday‘𝐴) = (bday‘𝐵)) ∧ (L‘𝐴) = (L‘𝐵)) → (R‘𝐴) = (R‘𝐵))
34	3, 33	oveq12d 7438	. . . . . 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 28295	. . . . . . 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 28295	. . . . . . 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 2803	. . . . 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 28116	. . 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 4029	. . 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 1569   ∈ wcel 2144   ⊆ 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  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:  ltonold  28652  onnolt  28657