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Theorem sltlpss 27236
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 27191 . . . . . . . . . . . 12 ( O ‘( bday 𝑋)) ⊆ No
21sseli 3940 . . . . . . . . . . 11 (𝑥 ∈ ( O ‘( bday 𝑋)) → 𝑥 No )
323ad2ant2 1134 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 No )
4 simp1l1 1266 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑋 No )
5 simp1l2 1267 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑌 No )
6 simp3 1138 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑋)
7 simp1r 1198 . . . . . . . . . 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 27107 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ 𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) → 𝑥 <s 𝑌)
983exp 1119 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday 𝑋)) → (𝑥 <s 𝑋𝑥 <s 𝑌)))
109imdistand 571 . . . . . . 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 6842 . . . . . . . . . . 11 (( bday 𝑋) = ( bday 𝑌) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
12113ad2ant3 1135 . . . . . . . . . 10 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
1312adantr 481 . . . . . . . . 9 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
1413eleq2d 2823 . . . . . . . 8 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( O ‘( bday 𝑋)) ↔ 𝑥 ∈ ( O ‘( bday 𝑌))))
1514anbi1d 630 . . . . . . 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 238 . . . . . 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 27193 . . . . . . . . 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 2823 . . . . . . 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 3427 . . . . . . 7 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋} ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋))
2119, 20bitrdi 286 . . . . . 6 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
22 leftval 27193 . . . . . . . . 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 2823 . . . . . . 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 3427 . . . . . . 7 (𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌} ↔ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌))
2624, 25bitrdi 286 . . . . . 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 293 . . . . 5 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (𝑥 ∈ ( L ‘𝑋) → 𝑥 ∈ ( L ‘𝑌)))
2827ssrdv 3950 . . . 4 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ( L ‘𝑋) ⊆ ( L ‘𝑌))
29 sltirr 27094 . . . . . . . . 9 (𝑌 No → ¬ 𝑌 <s 𝑌)
30293ad2ant2 1134 . . . . . . . 8 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ¬ 𝑌 <s 𝑌)
31 breq1 5108 . . . . . . . . 9 (𝑋 = 𝑌 → (𝑋 <s 𝑌𝑌 <s 𝑌))
3231notbid 317 . . . . . . . 8 (𝑋 = 𝑌 → (¬ 𝑋 <s 𝑌 ↔ ¬ 𝑌 <s 𝑌))
3330, 32syl5ibrcom 246 . . . . . . 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 407 . . . . 5 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ¬ 𝑋 = 𝑌)
36 simpr 485 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( L ‘𝑋) = ( L ‘𝑌))
37 lruneq 27235 . . . . . . . . . . 11 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
3837adantr 481 . . . . . . . . . 10 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
3938adantr 481 . . . . . . . . 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 4083 . . . . . . . 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 4240 . . . . . . . . . 10 ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
42 difid 4330 . . . . . . . . . . 11 (( L ‘𝑋) ∖ ( L ‘𝑋)) = ∅
4342uneq1i 4119 . . . . . . . . . 10 ((( L ‘𝑋) ∖ ( L ‘𝑋)) ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋)))
44 0un 4352 . . . . . . . . . 10 (∅ ∪ (( R ‘𝑋) ∖ ( L ‘𝑋))) = (( R ‘𝑋) ∖ ( L ‘𝑋))
4541, 43, 443eqtri 2768 . . . . . . . . 9 ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = (( R ‘𝑋) ∖ ( L ‘𝑋))
46 incom 4161 . . . . . . . . . . 11 (( L ‘𝑋) ∩ ( R ‘𝑋)) = (( R ‘𝑋) ∩ ( L ‘𝑋))
47 simpll1 1212 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 No )
48 lltropt 27202 . . . . . . . . . . . 12 (𝑋 No → ( L ‘𝑋) <<s ( R ‘𝑋))
49 ssltdisj 27160 . . . . . . . . . . . 12 (( L ‘𝑋) <<s ( R ‘𝑋) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
5047, 48, 493syl 18 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) ∩ ( R ‘𝑋)) = ∅)
5146, 50eqtr3id 2790 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅)
52 disjdif2 4439 . . . . . . . . . 10 ((( R ‘𝑋) ∩ ( L ‘𝑋)) = ∅ → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
5351, 52syl 17 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑋) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
5445, 53eqtrid 2788 . . . . . . . 8 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑋) ∪ ( R ‘𝑋)) ∖ ( L ‘𝑋)) = ( R ‘𝑋))
55 difundir 4240 . . . . . . . . . 10 ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
56 difid 4330 . . . . . . . . . . 11 (( L ‘𝑌) ∖ ( L ‘𝑌)) = ∅
5756uneq1i 4119 . . . . . . . . . 10 ((( L ‘𝑌) ∖ ( L ‘𝑌)) ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌)))
58 0un 4352 . . . . . . . . . 10 (∅ ∪ (( R ‘𝑌) ∖ ( L ‘𝑌))) = (( R ‘𝑌) ∖ ( L ‘𝑌))
5955, 57, 583eqtri 2768 . . . . . . . . 9 ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = (( R ‘𝑌) ∖ ( L ‘𝑌))
60 incom 4161 . . . . . . . . . . 11 (( L ‘𝑌) ∩ ( R ‘𝑌)) = (( R ‘𝑌) ∩ ( L ‘𝑌))
61 simpll2 1213 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑌 No )
62 lltropt 27202 . . . . . . . . . . . 12 (𝑌 No → ( L ‘𝑌) <<s ( R ‘𝑌))
63 ssltdisj 27160 . . . . . . . . . . . 12 (( L ‘𝑌) <<s ( R ‘𝑌) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
6461, 62, 633syl 18 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) ∩ ( R ‘𝑌)) = ∅)
6560, 64eqtr3id 2790 . . . . . . . . . 10 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅)
66 disjdif2 4439 . . . . . . . . . 10 ((( R ‘𝑌) ∩ ( L ‘𝑌)) = ∅ → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6765, 66syl 17 . . . . . . . . 9 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( R ‘𝑌) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6859, 67eqtrid 2788 . . . . . . . 8 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ((( L ‘𝑌) ∪ ( R ‘𝑌)) ∖ ( L ‘𝑌)) = ( R ‘𝑌))
6940, 54, 683eqtr3d 2784 . . . . . . 7 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → ( R ‘𝑋) = ( R ‘𝑌))
7036, 69oveq12d 7375 . . . . . 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)))
71 lrcut 27232 . . . . . . 7 (𝑋 No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
7247, 71syl 17 . . . . . 6 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
73 lrcut 27232 . . . . . . 7 (𝑌 No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
7461, 73syl 17 . . . . . 6 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌)
7570, 72, 743eqtr3d 2784 . . . . 5 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) ∧ ( L ‘𝑋) = ( L ‘𝑌)) → 𝑋 = 𝑌)
7635, 75mtand 814 . . . 4 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ 𝑋 <s 𝑌) → ¬ ( L ‘𝑋) = ( L ‘𝑌))
77 dfpss2 4045 . . . 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 413 . 2 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑋 <s 𝑌 → ( L ‘𝑋) ⊊ ( L ‘𝑌)))
80 dfpss3 4046 . . 3 (( L ‘𝑋) ⊊ ( L ‘𝑌) ↔ (( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)))
81 ssdif0 4323 . . . . . . 7 (( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) = ∅)
8281necon3bbii 2991 . . . . . 6 (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ (( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅)
83 n0 4306 . . . . . 6 ((( L ‘𝑌) ∖ ( L ‘𝑋)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
8482, 83bitri 274 . . . . 5 (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) ↔ ∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)))
85 eldif 3920 . . . . . . 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 2823 . . . . . . . . . . 11 (𝑌 No → (𝑥 ∈ ( L ‘𝑌) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑌)) ∣ 𝑥 <s 𝑌}))
8887, 25bitrdi 286 . . . . . . . . . 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 2823 . . . . . . . . . . . . 13 (𝑋 No → (𝑥 ∈ ( L ‘𝑋) ↔ 𝑥 ∈ {𝑥 ∈ ( O ‘( bday 𝑋)) ∣ 𝑥 <s 𝑋}))
9190, 20bitrdi 286 . . . . . . . . . . . 12 (𝑋 No → (𝑥 ∈ ( L ‘𝑋) ↔ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
9291notbid 317 . . . . . . . . . . 11 (𝑋 No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ ¬ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋)))
93 ianor 980 . . . . . . . . . . 11 (¬ (𝑥 ∈ ( O ‘( bday 𝑋)) ∧ 𝑥 <s 𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋))
9492, 93bitrdi 286 . . . . . . . . . 10 (𝑋 No → (¬ 𝑥 ∈ ( L ‘𝑋) ↔ (¬ 𝑥 ∈ ( O ‘( bday 𝑋)) ∨ ¬ 𝑥 <s 𝑋)))
9588, 94bi2anan9r 638 . . . . . . . . 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 1132 . . . . . . . 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 769 . . . . . . . . . . . 12 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 ∈ ( O ‘( bday 𝑌)))
98 simpl3 1193 . . . . . . . . . . . . 13 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → ( bday 𝑋) = ( bday 𝑌))
9998fveq2d 6846 . . . . . . . . . . . 12 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → ( O ‘( bday 𝑋)) = ( O ‘( bday 𝑌)))
10097, 99eleqtrrd 2841 . . . . . . . . . . 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 1212 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 No )
103 oldssno 27191 . . . . . . . . . . . . . 14 ( O ‘( bday 𝑌)) ⊆ No
104103, 97sselid 3942 . . . . . . . . . . . . 13 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑥 No )
105104adantr 481 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑥 No )
106 simpll2 1213 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑌 No )
107 simpl1 1191 . . . . . . . . . . . . . 14 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → 𝑋 No )
108 slenlt 27100 . . . . . . . . . . . . . 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 478 . . . . . . . . . . . 12 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 ≤s 𝑥)
111 simplrr 776 . . . . . . . . . . . 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 27109 . . . . . . . . . . 11 ((((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) ∧ ¬ 𝑥 <s 𝑋) → 𝑋 <s 𝑌)
113112ex 413 . . . . . . . . . 10 (((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) ∧ (𝑥 ∈ ( O ‘( bday 𝑌)) ∧ 𝑥 <s 𝑌)) → (¬ 𝑥 <s 𝑋𝑋 <s 𝑌))
114101, 113jaod 857 . . . . . . . . 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 454 . . . . . . . 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 239 . . . . . . 7 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ((𝑥 ∈ ( L ‘𝑌) ∧ ¬ 𝑥 ∈ ( L ‘𝑋)) → 𝑋 <s 𝑌))
11785, 116biimtrid 241 . . . . . 6 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
118117exlimdv 1936 . . . . 5 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (∃𝑥 𝑥 ∈ (( L ‘𝑌) ∖ ( L ‘𝑋)) → 𝑋 <s 𝑌))
11984, 118biimtrid 241 . . . 4 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (¬ ( L ‘𝑌) ⊆ ( L ‘𝑋) → 𝑋 <s 𝑌))
120119adantld 491 . . 3 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → ((( L ‘𝑋) ⊆ ( L ‘𝑌) ∧ ¬ ( L ‘𝑌) ⊆ ( L ‘𝑋)) → 𝑋 <s 𝑌))
12180, 120biimtrid 241 . 2 ((𝑋 No 𝑌 No ∧ ( bday 𝑋) = ( bday 𝑌)) → (( L ‘𝑋) ⊊ ( L ‘𝑌) → 𝑋 <s 𝑌))
12279, 121impbid 211 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  {crab 3407  cdif 3907  cun 3908  cin 3909  wss 3910  wpss 3911  c0 4282   class class class wbr 5105  cfv 6496  (class class class)co 7357   No csur 26988   <s cslt 26989   bday cbday 26990   ≤s csle 27092   <<s csslt 27120   |s cscut 27122   O cold 27173   L cleft 27175   R cright 27176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-1o 8412  df-2o 8413  df-no 26991  df-slt 26992  df-bday 26993  df-sle 27093  df-sslt 27121  df-scut 27123  df-made 27177  df-old 27178  df-left 27180  df-right 27181
This theorem is referenced by: (None)
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