Theorem 0nelxp 5695

Description: The empty set is not a member of a Cartesian product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof shortened by JJ, 13-Aug-2021.)

Assertion

Ref	Expression
0nelxp	⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

Proof of Theorem 0nelxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5456	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4	3	nesymi 3015	. . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩
5	4	intnanr 492	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	5	nex 1830	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7	6	nex 1830	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8		elxp 5684	. 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	7, 8	mtbir 326	1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∅c0 4286  ⟨cop 4595   × cxp 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-xp 5667

This theorem is referenced by:  0nelrel0  5721  nrelvOLD  5787  dmsn0  6210  onxpdisj  6488  mpoxopx0ov0  8208  dmtpos  8230  0nnq  10904  adderpq  10936  mulerpq  10937  lterpq  10950  0ncn  11113  structcnvcnv  17208  vtxval0  29389  iedgval0  29390  msrrcl  36035  oppfrcl2  49907  eloppf  49911