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Mirrors > Home > MPE Home > Th. List > Mathboxes > sltn0 | Structured version Visualization version GIF version |
Description: If 𝑋 is less than 𝑌, then either ( L ‘𝑌) or ( R ‘𝑋) is non-empty. (Contributed by Scott Fenton, 10-Dec-2024.) |
Ref | Expression |
---|---|
sltn0 | ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lltropt 33742 | . . . . 5 ⊢ (𝑋 ∈ No → ( L ‘𝑋) <<s ( R ‘𝑋)) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → ( L ‘𝑋) <<s ( R ‘𝑋)) |
3 | lltropt 33742 | . . . . 5 ⊢ (𝑌 ∈ No → ( L ‘𝑌) <<s ( R ‘𝑌)) | |
4 | 3 | adantl 485 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → ( L ‘𝑌) <<s ( R ‘𝑌)) |
5 | lrcut 33769 | . . . . . 6 ⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋) | |
6 | 5 | eqcomd 2742 | . . . . 5 ⊢ (𝑋 ∈ No → 𝑋 = (( L ‘𝑋) |s ( R ‘𝑋))) |
7 | 6 | adantr 484 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → 𝑋 = (( L ‘𝑋) |s ( R ‘𝑋))) |
8 | lrcut 33769 | . . . . . 6 ⊢ (𝑌 ∈ No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌) | |
9 | 8 | eqcomd 2742 | . . . . 5 ⊢ (𝑌 ∈ No → 𝑌 = (( L ‘𝑌) |s ( R ‘𝑌))) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → 𝑌 = (( L ‘𝑌) |s ( R ‘𝑌))) |
11 | sltrec 33700 | . . . 4 ⊢ (((( L ‘𝑋) <<s ( R ‘𝑋) ∧ ( L ‘𝑌) <<s ( R ‘𝑌)) ∧ (𝑋 = (( L ‘𝑋) |s ( R ‘𝑋)) ∧ 𝑌 = (( L ‘𝑌) |s ( R ‘𝑌)))) → (𝑋 <s 𝑌 ↔ (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 ∨ ∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌))) | |
12 | 2, 4, 7, 10, 11 | syl22anc 839 | . . 3 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → (𝑋 <s 𝑌 ↔ (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 ∨ ∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌))) |
13 | 12 | biimp3a 1471 | . 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 ∨ ∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌)) |
14 | rexn0 4408 | . . 3 ⊢ (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 → ( L ‘𝑌) ≠ ∅) | |
15 | rexn0 4408 | . . 3 ⊢ (∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌 → ( R ‘𝑋) ≠ ∅) | |
16 | 14, 15 | orim12i 909 | . 2 ⊢ ((∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 ∨ ∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅)) |
17 | 13, 16 | syl 17 | 1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (( L ‘𝑌) ≠ ∅ ∨ ( R ‘𝑋) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 No csur 33529 <s cslt 33530 ≤s csle 33633 <<s csslt 33661 |s cscut 33663 L cleft 33715 R cright 33716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-wrecs 8025 df-recs 8086 df-1o 8180 df-2o 8181 df-no 33532 df-slt 33533 df-bday 33534 df-sle 33634 df-sslt 33662 df-scut 33664 df-made 33717 df-old 33718 df-left 33720 df-right 33721 |
This theorem is referenced by: (None) |
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