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| Mirrors > Home > MPE Home > Th. List > ssltdisj | Structured version Visualization version GIF version | ||
| Description: If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| ssltdisj | ⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssltss1 27833 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
| 2 | 1 | sselda 3983 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No ) | 
| 3 | sltirr 27791 | . . . . 5 ⊢ (𝑥 ∈ No → ¬ 𝑥 <s 𝑥) | |
| 4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 <s 𝑥) | 
| 5 | ssltsepc 27838 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 <s 𝑥) | |
| 6 | 5 | 3expa 1119 | . . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ∈ 𝐵) → 𝑥 <s 𝑥) | 
| 7 | 4, 6 | mtand 816 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵) | 
| 8 | 7 | ralrimiva 3146 | . 2 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | 
| 9 | disj 4450 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
| 10 | 8, 9 | sylibr 234 | 1 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 No csur 27684 <s cslt 27685 <<s csslt 27825 | 
| 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-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-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-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-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-sslt 27826 | 
| This theorem is referenced by: sltlpss 27945 slelss 27946 | 
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