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Mirrors > Home > MPE Home > Th. List > 2spot0 | Structured version Visualization version GIF version |
Description: If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.) |
Ref | Expression |
---|---|
2spot0 | ⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → (𝑉 2SPathsOt 𝐸) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
2 | eleq1 2676 | . . . 4 ⊢ (𝑉 = ∅ → (𝑉 ∈ V ↔ ∅ ∈ V)) | |
3 | 1, 2 | mpbiri 247 | . . 3 ⊢ (𝑉 = ∅ → 𝑉 ∈ V) |
4 | 2spthsot 26395 | . . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ 𝑋) → (𝑉 2SPathsOt 𝐸) = {𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)}) | |
5 | 3, 4 | sylan 487 | . 2 ⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → (𝑉 2SPathsOt 𝐸) = {𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)}) |
6 | 2spthonot3v 26403 | . . . . . . . . 9 ⊢ (𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))) | |
7 | n0i 3879 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝑉 → ¬ 𝑉 = ∅) | |
8 | 7 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ¬ 𝑉 = ∅) |
9 | 8 | 3ad2ant2 1076 | . . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) → ¬ 𝑉 = ∅) |
10 | 6, 9 | syl 17 | . . . . . . . 8 ⊢ (𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏) → ¬ 𝑉 = ∅) |
11 | 10 | con2i 133 | . . . . . . 7 ⊢ (𝑉 = ∅ → ¬ 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) |
12 | 11 | ad4antr 764 | . . . . . 6 ⊢ (((((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ¬ 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) |
13 | 12 | nrexdv 2984 | . . . . 5 ⊢ ((((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑎 ∈ 𝑉) → ¬ ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) |
14 | 13 | nrexdv 2984 | . . . 4 ⊢ (((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) → ¬ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) |
15 | 14 | ralrimiva 2949 | . . 3 ⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → ∀𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ¬ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) |
16 | rabeq0 3911 | . . 3 ⊢ ({𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)} = ∅ ↔ ∀𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ¬ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)) | |
17 | 15, 16 | sylibr 223 | . 2 ⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → {𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)} = ∅) |
18 | 5, 17 | eqtrd 2644 | 1 ⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → (𝑉 2SPathsOt 𝐸) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∅c0 3874 × cxp 5036 (class class class)co 6549 2SPathsOt c2spthot 26383 2SPathOnOt c2pthonot 26384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-2spthonot 26387 df-2spthsot 26388 |
This theorem is referenced by: (None) |
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