Theorem 2spot0 26591

Description: If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.)

Assertion

Ref	Expression
2spot0	⊢ ((𝑉 = ∅ ∧ 𝐸 ∈ 𝑋) → (𝑉 2SPathsOt 𝐸) = ∅)

Proof of Theorem 2spot0

Dummy variables 𝑎 𝑏 𝑝 are mutually distinct and distinct from all other variables.

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) = ∅)

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)