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Theorem sltlpss 27945
Description: If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
Assertion
Ref Expression
sltlpss ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))

Proof of Theorem sltlpss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oldssno 27900 . . . . . . . . . . . 12 ( O ‘( bday 𝑋)) ⊆ No
21sseli 3979 . . . . . . . . . . 11 (𝑥 ∈ ( O ‘( bday 𝑋)) → 𝑥 No )
323ad2ant2 1135 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 No )
4 simp1l1 1267 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 No )
5 simp1l2 1268 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑌 No )
6 simp3 1139 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑋)
7 simp1r 1199 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 <s 𝑌)
83, 4, 5, 6, 7slttrd 27804 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑌)
983exp 1120 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday 𝑋)) → (𝑥 <s 𝑋𝑥 <s 𝑌)))
109imdistand 570 . . . . . . 7 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑌)))
11 fveq2 6906 . . . . . . . . . . 11 (( bday 𝑋) = ( bday 𝑌) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
12113ad2ant3 1136 . . . . . . . . . 10 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
1312adantr 480 . . . . . . . . 9 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
1413eleq2d 2827 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ 𝑥 ∈ ( O ‘( bday 𝑌))))
1514anbi1d 631 . . . . . . 7 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑌) ↔ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)))
1610, 15sylibd 239 . . . . . 6 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ((𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)))
17 leftval 27902 . . . . . . . . 9 ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}
1817a1i 11 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋})
1918eleq2d 2827 . . . . . . 7 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
20 rabid 3458 . . . . . . 7 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
2119, 20bitrdi 287 . . . . . 6 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
22 leftval 27902 . . . . . . . . 9 ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌}
2322a1i 11 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌})
2423eleq2d 2827 . . . . . . 7 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌}))
25 rabid 3458 . . . . . . 7 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌} ↔ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌))
2624, 25bitrdi 287 . . . . . 6 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑌) ↔ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)))
2716, 21, 263imtr4d 294 . . . . 5 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) → 𝑥 ∈ ( L ‘𝑌)))
2827ssrdv 3989 . . . 4 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊆ ( L ‘𝑌))
29 sltirr 27791 . . . . . . . . 9 (𝑌 No → ¬ 𝑌 <s 𝑌)
30293ad2ant2 1135 . . . . . . . 8 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ¬ 𝑌 <s 𝑌)
31 breq1 5146 . . . . . . . . 9 (𝑋 = 𝑌 → (𝑋 <s 𝑌𝑌 <s 𝑌))
3231notbid 318 . . . . . . . 8 (𝑋 = 𝑌 → (¬ 𝑋 <s 𝑌 ↔ ¬ 𝑌 <s 𝑌))
3330, 32syl5ibrcom 247 . . . . . . 7 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 = 𝑌 → ¬ 𝑋 <s 𝑌))
3433con2d 134 . . . . . 6 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 <s 𝑌 → ¬ 𝑋 = 𝑌))
3534imp 406 . . . . 5 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ¬ 𝑋 = 𝑌)
36 simpr 484 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( L ‘𝑋) = ( L ‘𝑌))
37 lruneq 27944 . . . . . . . . . . 11 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
3837adantr 480 . . . . . . . . . 10 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
3938adantr 480 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
4039, 36difeq12d 4127 . . . . . . . 8 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)))
41 difundir 4291 . . . . . . . . . 10 ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
42 difid 4376 . . . . . . . . . . 11 (( L ‘𝑋) ∖ ( L ‘𝑋)) = ∅
4342uneq1i 4164 . . . . . . . . . 10 ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
44 0un 4396 . . . . . . . . . 10 (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (( R ‘𝑋) ∖ ( L ‘𝑋))
4541, 43, 443eqtri 2769 . . . . . . . . 9 ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = (( R ‘𝑋) ∖ ( L ‘𝑋))
46 incom 4209 . . . . . . . . . . 11 (( L ‘𝑋) ∩ ( R ‘𝑋)) = (( R ‘𝑋) ∩ ( L ‘𝑋))
47 lltropt 27911 . . . . . . . . . . . 12 ( L ‘𝑋) <<s ( R ‘𝑋)
48 ssltdisj 27866 . . . . . . . . . . . 12 (( L ‘𝑋) <<s ( R ‘𝑋) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
4947, 48mp1i 13 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
5046, 49eqtr3id 2791 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅)
51 disjdif2 4480 . . . . . . . . . 10 ((( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅ → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
5250, 51syl 17 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
5345, 52eqtrid 2789 . . . . . . . 8 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
54 difundir 4291 . . . . . . . . . 10 ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
55 difid 4376 . . . . . . . . . . 11 (( L ‘𝑌) ∖ ( L ‘𝑌)) = ∅
5655uneq1i 4164 . . . . . . . . . 10 ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
57 0un 4396 . . . . . . . . . 10 (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (( R ‘𝑌) ∖ ( L ‘𝑌))
5854, 56, 573eqtri 2769 . . . . . . . . 9 ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = (( R ‘𝑌) ∖ ( L ‘𝑌))
59 incom 4209 . . . . . . . . . . 11 (( L ‘𝑌) ∩ ( R ‘𝑌)) = (( R ‘𝑌) ∩ ( L ‘𝑌))
60 lltropt 27911 . . . . . . . . . . . 12 ( L ‘𝑌) <<s ( R ‘𝑌)
61 ssltdisj 27866 . . . . . . . . . . . 12 (( L ‘𝑌) <<s ( R ‘𝑌) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
6260, 61mp1i 13 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
6359, 62eqtr3id 2791 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅)
64 disjdif2 4480 . . . . . . . . . 10 ((( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅ → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6563, 64syl 17 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6658, 65eqtrid 2789 . . . . . . . 8 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6740, 53, 663eqtr3d 2785 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( R ‘𝑋) = ( R ‘𝑌))
6836, 67oveq12d 7449 . . . . . 6 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑌) |s ( R ‘𝑌)))
69 simpll1 1213 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 No )
70 lrcut 27941 . . . . . . 7 (𝑋 No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
7169, 70syl 17 . . . . . 6 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
72 simpll2 1214 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑌 No )
73 lrcut 27941 . . . . . . 7 (𝑌 No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
7472, 73syl 17 . . . . . 6 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
7568, 71, 743eqtr3d 2785 . . . . 5 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 = 𝑌)
7635, 75mtand 816 . . . 4 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ¬ ( L ‘𝑋) = ( L ‘𝑌))
77 dfpss2 4088 . . . 4 (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑋) = ( L ‘𝑌)))
7828, 76, 77sylanbrc 583 . . 3 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊊ ( L ‘𝑌))
7978ex 412 . 2 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 <s 𝑌 → ( L ‘𝑋) ⊊ ( L ‘𝑌)))
80 dfpss3 4089 . . 3 (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)))
81 ssdif0 4366 . . . . . . 7 (( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) = ∅)
8281necon3bbii 2988 . . . . . 6 (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅)
83 n0 4353 . . . . . 6 ((( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
8482, 83bitri 275 . . . . 5 (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
85 eldif 3961 . . . . . . 7 (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) ↔ (𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)))
8622a1i 11 . . . . . . . . . . . 12 (𝑌 No → ( L ‘𝑌) = {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌})
8786eleq2d 2827 . . . . . . . . . . 11 (𝑌 No → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌}))
8887, 25bitrdi 287 . . . . . . . . . 10 (𝑌 No → (𝑥 ∈ ( L ‘𝑌) ↔ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)))
8917a1i 11 . . . . . . . . . . . . . 14 (𝑋 No → ( L ‘𝑋) = {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋})
9089eleq2d 2827 . . . . . . . . . . . . 13 (𝑋 No → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
9190, 20bitrdi 287 . . . . . . . . . . . 12 (𝑋 No → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
9291notbid 318 . . . . . . . . . . 11 (𝑋 No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ ¬ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
93 ianor 984 . . . . . . . . . . 11 (¬ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋))
9492, 93bitrdi 287 . . . . . . . . . 10 (𝑋 No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋)))
9588, 94bi2anan9r 639 . . . . . . . . 9 ((𝑋 No 𝑌 No ) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) ↔ ((𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋))))
96953adant3 1133 . . . . . . . 8 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) ↔ ((𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋))))
97 simprl 771 . . . . . . . . . . . 12 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday 𝑌)))
98 simpl3 1194 . . . . . . . . . . . . 13 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → ( bday 𝑋) = ( bday 𝑌))
9998fveq2d 6910 . . . . . . . . . . . 12 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
10097, 99eleqtrrd 2844 . . . . . . . . . . 11 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday 𝑋)))
101100pm2.24d 151 . . . . . . . . . 10 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) → 𝑋 <s 𝑌))
102 simpll1 1213 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 No )
103 oldssno 27900 . . . . . . . . . . . . . 14 ( O ‘( bday 𝑌)) ⊆ No
104103, 97sselid 3981 . . . . . . . . . . . . 13 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 No )
105104adantr 480 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 No )
106 simpll2 1214 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑌 No )
107 simpl1 1192 . . . . . . . . . . . . . 14 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑋 No )
108 slenlt 27797 . . . . . . . . . . . . . 14 ((𝑋 No 𝑥 No ) → (𝑋 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑋))
109107, 104, 108syl2anc 584 . . . . . . . . . . . . 13 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → (𝑋 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑋))
110109biimpar 477 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ≤s 𝑥)
111 simplrr 778 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 <s 𝑌)
112102, 105, 106, 110, 111slelttrd 27806 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 <s 𝑌)
113112ex 412 . . . . . . . . . 10 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → (¬ 𝑥 <s 𝑋𝑋 <s 𝑌))
114101, 113jaod 860 . . . . . . . . 9 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → ((¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋) → 𝑋 <s 𝑌))
115114expimpd 453 . . . . . . . 8 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (((𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌) ∧ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋)) → 𝑋 <s 𝑌))
11696, 115sylbid 240 . . . . . . 7 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 <s 𝑌))
11785, 116biimtrid 242 . . . . . 6 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
118117exlimdv 1933 . . . . 5 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
11984, 118biimtrid 242 . . . 4 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) → 𝑋 <s 𝑌))
120119adantld 490 . . 3 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ((( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)) → 𝑋 <s 𝑌))
12180, 120biimtrid 242 . 2 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (( L ‘𝑋) ⊊ ( L ‘𝑌) → 𝑋 <s 𝑌))
12279, 121impbid 212 1 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  {crab 3436  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   O cold 27882   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:  slelss  27946
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