Theorem 0nelelxp 5713

Description: A member of a Cartesian product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)

Assertion

Ref	Expression
0nelelxp	⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Proof of Theorem 0nelelxp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5701	. 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		0nelop 5498	. . . . 5 ⊢ ¬ ∅ ∈ ⟨𝑥, 𝑦⟩
3		eleq2 2814	. . . . 5 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
4	2, 3	mtbiri 326	. . . 4 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ 𝐶)
5	4	adantr 479	. . 3 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
6	5	exlimivv 1927	. 2 ⊢ (∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
7	1, 6	sylbi 216	1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∅c0 4322  ⟨cop 4636   × cxp 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5212  df-xp 5684

This theorem is referenced by:  dmsn0el  6217  onxpdisj  6497