Theorem slelss 27961

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 27960	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 <s 𝐵 ↔ ( L ‘𝐴) ⊊ ( L ‘𝐵)))
2		fveq2 6907	. . . 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 27959	. . . . . . . . . 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 4137	. . . . . . . 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 4297	. . . . . . . . . 10 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
8		difid 4382	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∖ ( L ‘𝐴)) = ∅
9	8	uneq1i 4174	. . . . . . . . . 10 ⊢ ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
10		0un 4402	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (( R ‘𝐴) ∖ ( L ‘𝐴))
11	7, 9, 10	3eqtri 2767	. . . . . . . . 9 ⊢ ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = (( R ‘𝐴) ∖ ( L ‘𝐴))
12		incom 4217	. . . . . . . . . . 11 ⊢ (( L ‘𝐴) ∩ ( R ‘𝐴)) = (( R ‘𝐴) ∩ ( L ‘𝐴))
13		lltropt 27926	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
14		ssltdisj 27881	. . . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐴) ∩ ( L ‘𝐴)) = ∅)
17		disjdif2 4486	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
20		difundir 4297	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
21		difid 4382	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∖ ( L ‘𝐵)) = ∅
22	21	uneq1i 4174	. . . . . . . . . 10 ⊢ ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
23		0un 4402	. . . . . . . . . 10 ⊢ (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (( R ‘𝐵) ∖ ( L ‘𝐵))
24	20, 22, 23	3eqtri 2767	. . . . . . . . 9 ⊢ ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = (( R ‘𝐵) ∖ ( L ‘𝐵))
25		incom 4217	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∩ ( R ‘𝐵)) = (( R ‘𝐵) ∩ ( L ‘𝐵))
26		lltropt 27926	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
27		ssltdisj 27881	. . . . . . . . . . . 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 2789	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐵) ∩ ( L ‘𝐵)) = ∅)
30		disjdif2 4486	. . . . . . . . . 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 2787	. . . . . . . 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 2783	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ( R ‘𝐴) = ( R ‘𝐵))
34	3, 33	oveq12d 7449	. . . . . 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 1190	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐴 ∈ No )
36		lrcut 27956	. . . . . . 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 1191	. . . . . . 7 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐵 ∈ No )
39		lrcut 27956	. . . . . . 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 2783	. . . . 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 27814	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
46	45	3adant3 1131	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ ( bday ‘𝐴) = ( bday ‘𝐵)) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))
47		sspss 4112	. . 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 1537   ∈ wcel 2106   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963   ⊊ wpss 3964  ∅c0 4339   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431   No csur 27699   <s cslt 27700   bday cbday 27701   ≤s csle 27804   <<s csslt 27840   |s cscut 27842   L cleft 27899   R cright 27900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905

This theorem is referenced by:  sltonold  28298