Theorem 0nelelxp 5592

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 5580	. 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		0nelop 5388	. . . . 5 ⊢ ¬ ∅ ∈ ⟨𝑥, 𝑦⟩
3		eleq2 2903	. . . . 5 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
4	2, 3	mtbiri 329	. . . 4 ⊢ (𝐶 = ⟨𝑥, 𝑦⟩ → ¬ ∅ ∈ 𝐶)
5	4	adantr 483	. . 3 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
6	5	exlimivv 1933	. 2 ⊢ (∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
7	1, 6	sylbi 219	1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∅c0 4293  ⟨cop 4575   × cxp 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-opab 5131  df-xp 5563

This theorem is referenced by:  dmsn0el  6070  onxpdisj  6312