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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33746 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
2 | 1 | sselda 3916 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No ) |
3 | sltirr 33712 | . . . . 5 ⊢ (𝑥 ∈ No → ¬ 𝑥 <s 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 <s 𝑥) |
5 | ssltsepc 33750 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 <s 𝑥) | |
6 | 5 | 3expa 1120 | . . . 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 3106 | . 2 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
9 | disj 4377 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
10 | 8, 9 | sylibr 237 | 1 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∩ cin 3880 ∅c0 4252 class class class wbr 5068 No csur 33606 <s cslt 33607 <<s csslt 33738 |
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 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
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 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-ord 6234 df-on 6235 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-fv 6406 df-1o 8223 df-2o 8224 df-no 33609 df-slt 33610 df-sslt 33739 |
This theorem is referenced by: sltlpss 33850 |
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