Theorem 0nelxp 2475

Description: The empty set is not a member of a cross product.

Assertion

Ref	Expression
0nelxp	⊢ ¬ ∅ ∈ (A × B)

Proof of Theorem 0nelxp

Step	Hyp	Ref	Expression
1		noel 1711	. . . . . 6 ⊢ ¬ {x} ∈ ∅
2		opi1 1895	. . . . . . 7 ⊢ {x} ∈ ⟨x, y⟩
3		eleq2 1150	. . . . . . 7 ⊢ (∅ = ⟨x, y⟩ → ({x} ∈ ∅ ↔ {x} ∈ ⟨x, y⟩))
4	2, 3	mpbiri 169	. . . . . 6 ⊢ (∅ = ⟨x, y⟩ → {x} ∈ ∅)
5	1, 4	mto 93	. . . . 5 ⊢ ¬ ∅ = ⟨x, y⟩
6	5	intnanr 517	. . . 4 ⊢ ¬ (∅ = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B))
7	6	nex 779	. . 3 ⊢ ¬ ∃y(∅ = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B))
8	7	nex 779	. 2 ⊢ ¬ ∃x∃y(∅ = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B))
9		elxp 2442	. 2 ⊢ (∅ ∈ (A × B) ↔ ∃x∃y(∅ = ⟨x, y⟩ ∧ (x ∈ A ∧ y ∈ B)))
10	8, 9	mtbir 167	1 ⊢ ¬ ∅ ∈ (A × B)

Syntax hints:  ¬ wn 1   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {csn 1808  ⟨cop 1810   × cxp 2408

This theorem is referenced by:  onxpdisj 2476  nfunv 2693  0ncn 4045

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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