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Mirrors > Home > MPE Home > Th. List > 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 27926 | . . . . 5 ⊢ ( L ‘𝑋) <<s ( R ‘𝑋) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → ( L ‘𝑋) <<s ( R ‘𝑋)) |
3 | lltropt 27926 | . . . . 5 ⊢ ( L ‘𝑌) <<s ( R ‘𝑌) | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → ( L ‘𝑌) <<s ( R ‘𝑌)) |
5 | lrcut 27956 | . . . . . 6 ⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋) | |
6 | 5 | eqcomd 2741 | . . . . 5 ⊢ (𝑋 ∈ No → 𝑋 = (( L ‘𝑋) |s ( R ‘𝑋))) |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → 𝑋 = (( L ‘𝑋) |s ( R ‘𝑋))) |
8 | lrcut 27956 | . . . . . 6 ⊢ (𝑌 ∈ No → (( L ‘𝑌) |s ( R ‘𝑌)) = 𝑌) | |
9 | 8 | eqcomd 2741 | . . . . 5 ⊢ (𝑌 ∈ No → 𝑌 = (( L ‘𝑌) |s ( R ‘𝑌))) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ) → 𝑌 = (( L ‘𝑌) |s ( R ‘𝑌))) |
11 | sltrec 27880 | . . . 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 1468 | . 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑋 <s 𝑌) → (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 ∨ ∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌)) |
14 | rexn0 4517 | . . 3 ⊢ (∃𝑦 ∈ ( L ‘𝑌)𝑋 ≤s 𝑦 → ( L ‘𝑌) ≠ ∅) | |
15 | rexn0 4517 | . . 3 ⊢ (∃𝑥 ∈ ( R ‘𝑋)𝑥 ≤s 𝑌 → ( R ‘𝑋) ≠ ∅) | |
16 | 14, 15 | orim12i 908 | . 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∅c0 4339 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 No csur 27699 <s cslt 27700 ≤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: (None) |
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