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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 27848 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
2 | 1 | sselda 3995 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No ) |
3 | sltirr 27806 | . . . . 5 ⊢ (𝑥 ∈ No → ¬ 𝑥 <s 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 <s 𝑥) |
5 | ssltsepc 27853 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 <s 𝑥) | |
6 | 5 | 3expa 1117 | . . . 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 3144 | . 2 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
9 | disj 4456 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ∅c0 4339 class class class wbr 5148 No csur 27699 <s cslt 27700 <<s csslt 27840 |
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-sep 5302 ax-nul 5312 ax-pr 5438 |
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-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-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-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-1o 8505 df-2o 8506 df-no 27702 df-slt 27703 df-sslt 27841 |
This theorem is referenced by: sltlpss 27960 slelss 27961 |
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