Theorem 0nelxp 5685

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 3455	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3455	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5443	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4	3	nesymi 3013	. . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩
5	4	intnanr 493	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	5	nex 1833	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7	6	nex 1833	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8		elxp 5674	. 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	7, 8	mtbir 326	1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∅c0 4279  ⟨cop 4590   × cxp 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5657

This theorem is used by:  0nelrel0  5711  nrelvOLD  5778  dmsn0  6210  onxpdisj  6490  mpoxopx0ov0  8233  dmtpos  8255  0nnq  11009  adderpq  11041  mulerpq  11042  lterpq  11055  0ncn  11218  structcnvcnv  17331  vtxval0  29617  iedgval0  29618  msrrcl  36308  oppfrcl2  50236  eloppf  50240