Theorem 0nelxp 5652

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 3435	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3435	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 5414	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4	3	nesymi 2991	. . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩
5	4	intnanr 488	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6	5	nex 1807	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7	6	nex 1807	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8		elxp 5641	. 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
9	7, 8	mtbir 324	1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∅c0 4261  ⟨cop 4561   × cxp 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5135  df-xp 5624

This theorem is referenced by:  0nelrel0  5678  nrelv  5743  dmsn0  6160  onxpdisj  6437  mpoxopx0ov0  8156  dmtpos  8178  0nnq  10838  adderpq  10870  mulerpq  10871  lterpq  10884  0ncn  11047  structcnvcnv  17114  vtxval0  29126  iedgval0  29127  msrrcl  35771  oppfrcl2  49619  eloppf  49623