Theorem 0nelxp 4390

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 2604	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 2604	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	opnzi 3990	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
4		simpl 107	. . . . . . 7 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
5	4	eqcomd 2086	. . . . . 6 ⊢ ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
6	5	necon3ai 2294	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
7	3, 6	ax-mp 7	. . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
8	7	nex 1429	. . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
9	8	nex 1429	. 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
10		elxp 4380	. 2 ⊢ (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
11	9, 10	mtbir 628	1 ⊢ ¬ ∅ ∈ (𝐴 × 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ≠ wne 2245  ∅c0 3251  ⟨cop 3401   × cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369

This theorem is referenced by:  dmsn0  4808  nfunv  4953  reldmtpos  5891  dmtpos  5894  0ncn  7000