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Theorem slelss 27946
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
StepHypRef Expression
1 sltlpss 27945 . . 3 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 <s 𝐵 ↔ ( L ‘𝐴) ⊊ ( L ‘𝐵)))
2 fveq2 6906 . . . 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 27944 . . . . . . . . . 10 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
54adantr 480 . . . . . . . . 9 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (( L ‘𝐵) ∪ ( R ‘𝐵)))
65, 3difeq12d 4127 . . . . . . . 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 4291 . . . . . . . . . 10 ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
8 difid 4376 . . . . . . . . . . 11 (( L ‘𝐴) ∖ ( L ‘𝐴)) = ∅
98uneq1i 4164 . . . . . . . . . 10 ((( L ‘𝐴) ∖ ( L ‘𝐴)) ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴)))
10 0un 4396 . . . . . . . . . 10 (∅ ∪ (( R ‘𝐴) ∖ ( L ‘𝐴))) = (( R ‘𝐴) ∖ ( L ‘𝐴))
117, 9, 103eqtri 2769 . . . . . . . . 9 ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = (( R ‘𝐴) ∖ ( L ‘𝐴))
12 incom 4209 . . . . . . . . . . 11 (( L ‘𝐴) ∩ ( R ‘𝐴)) = (( R ‘𝐴) ∩ ( L ‘𝐴))
13 lltropt 27911 . . . . . . . . . . . 12 ( L ‘𝐴) <<s ( R ‘𝐴)
14 ssltdisj 27866 . . . . . . . . . . . 12 (( L ‘𝐴) <<s ( R ‘𝐴) → (( L ‘𝐴) ∩ ( R ‘𝐴)) = ∅)
1513, 14mp1i 13 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐴) ∩ ( R ‘𝐴)) = ∅)
1612, 15eqtr3id 2791 . . . . . . . . . 10 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐴) ∩ ( L ‘𝐴)) = ∅)
17 disjdif2 4480 . . . . . . . . . 10 ((( R ‘𝐴) ∩ ( L ‘𝐴)) = ∅ → (( R ‘𝐴) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
1816, 17syl 17 . . . . . . . . 9 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐴) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
1911, 18eqtrid 2789 . . . . . . . 8 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ((( L ‘𝐴) ∪ ( R ‘𝐴)) ∖ ( L ‘𝐴)) = ( R ‘𝐴))
20 difundir 4291 . . . . . . . . . 10 ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
21 difid 4376 . . . . . . . . . . 11 (( L ‘𝐵) ∖ ( L ‘𝐵)) = ∅
2221uneq1i 4164 . . . . . . . . . 10 ((( L ‘𝐵) ∖ ( L ‘𝐵)) ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵)))
23 0un 4396 . . . . . . . . . 10 (∅ ∪ (( R ‘𝐵) ∖ ( L ‘𝐵))) = (( R ‘𝐵) ∖ ( L ‘𝐵))
2420, 22, 233eqtri 2769 . . . . . . . . 9 ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = (( R ‘𝐵) ∖ ( L ‘𝐵))
25 incom 4209 . . . . . . . . . . 11 (( L ‘𝐵) ∩ ( R ‘𝐵)) = (( R ‘𝐵) ∩ ( L ‘𝐵))
26 lltropt 27911 . . . . . . . . . . . 12 ( L ‘𝐵) <<s ( R ‘𝐵)
27 ssltdisj 27866 . . . . . . . . . . . 12 (( L ‘𝐵) <<s ( R ‘𝐵) → (( L ‘𝐵) ∩ ( R ‘𝐵)) = ∅)
2826, 27mp1i 13 . . . . . . . . . . 11 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐵) ∩ ( R ‘𝐵)) = ∅)
2925, 28eqtr3id 2791 . . . . . . . . . 10 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐵) ∩ ( L ‘𝐵)) = ∅)
30 disjdif2 4480 . . . . . . . . . 10 ((( R ‘𝐵) ∩ ( L ‘𝐵)) = ∅ → (( R ‘𝐵) ∖ ( L ‘𝐵)) = ( R ‘𝐵))
3129, 30syl 17 . . . . . . . . 9 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( R ‘𝐵) ∖ ( L ‘𝐵)) = ( R ‘𝐵))
3224, 31eqtrid 2789 . . . . . . . 8 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∖ ( L ‘𝐵)) = ( R ‘𝐵))
336, 19, 323eqtr3d 2785 . . . . . . 7 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → ( R ‘𝐴) = ( R ‘𝐵))
343, 33oveq12d 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 1192 . . . . . . 7 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐴 No )
36 lrcut 27941 . . . . . . 7 (𝐴 No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
3735, 36syl 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 27941 . . . . . . 7 (𝐵 No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
4038, 39syl 17 . . . . . 6 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
4134, 37, 403eqtr3d 2785 . . . . 5 (((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) ∧ ( L ‘𝐴) = ( L ‘𝐵)) → 𝐴 = 𝐵)
4241ex 412 . . . 4 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (( L ‘𝐴) = ( L ‘𝐵) → 𝐴 = 𝐵))
432, 42impbid2 226 . . 3 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 = 𝐵 ↔ ( L ‘𝐴) = ( L ‘𝐵)))
441, 43orbi12d 919 . 2 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → ((𝐴 <s 𝐵𝐴 = 𝐵) ↔ (( L ‘𝐴) ⊊ ( L ‘𝐵) ∨ ( L ‘𝐴) = ( L ‘𝐵))))
45 sleloe 27799 . . 3 ((𝐴 No 𝐵 No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))
46453adant3 1133 . 2 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵𝐴 = 𝐵)))
47 sspss 4102 . . 3 (( L ‘𝐴) ⊆ ( L ‘𝐵) ↔ (( L ‘𝐴) ⊊ ( L ‘𝐵) ∨ ( L ‘𝐴) = ( L ‘𝐵)))
4847a1i 11 . 2 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (( L ‘𝐴) ⊆ ( L ‘𝐵) ↔ (( L ‘𝐴) ⊊ ( L ‘𝐵) ∨ ( L ‘𝐴) = ( L ‘𝐵))))
4944, 46, 483bitr4d 311 1 ((𝐴 No 𝐵 No ∧ ( bday 𝐴) = ( bday 𝐵)) → (𝐴 ≤s 𝐵 ↔ ( L ‘𝐴) ⊆ ( L ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  cdif 3948  cun 3949  cin 3950  wss 3951  wpss 3952  c0 4333   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   bday cbday 27686   ≤s csle 27789   <<s csslt 27825   |s cscut 27827   L cleft 27884   R cright 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-left 27889  df-right 27890
This theorem is referenced by:  sltonold  28283
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