Theorem 0nelxp 4759

Description: The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

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 2806	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2806	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 4333	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4		simpl 109	. . . . . . 7 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
5	4	eqcomd 2237	. . . . . 6 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
6	5	necon3ai 2452	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	3, 6	ax-mp 5	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	7	nex 1549	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9	8	nex 1549	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
10		elxp 4748	. 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
11	9, 10	mtbir 678	1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202   ≠ wne 2403  ∅c0 3496  ⟨cop 3676   × cxp 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737

This theorem is referenced by:  0nelrel  4778  dmsn0  5211  nfunv  5366  reldmtpos  6462  dmtpos  6465  0ncn  8094  structcnvcnv  13161  vtxval0  15977  iedgval0  15978