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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 27150 | . . . . . 6 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | |
2 | 1 | sselda 3945 | . . . . 5 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No ) |
3 | sltirr 27110 | . . . . 5 ⊢ (𝑥 ∈ No → ¬ 𝑥 <s 𝑥) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 <s 𝑥) |
5 | ssltsepc 27154 | . . . . 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 815 | . . 3 ⊢ ((𝐴 <<s 𝐵 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵) |
8 | 7 | ralrimiva 3140 | . 2 ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) |
9 | disj 4408 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) | |
10 | 8, 9 | sylibr 233 | 1 ⊢ (𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∩ cin 3910 ∅c0 4283 class class class wbr 5106 No csur 27004 <s cslt 27005 <<s csslt 27142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-1o 8413 df-2o 8414 df-no 27007 df-slt 27008 df-sslt 27143 |
This theorem is referenced by: sltlpss 27258 |
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