Theorem 0nelelxp 4754

Description: A member of a cross 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 4742	. 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
2		0nelop 4340	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑥, 𝑦⟩
3		simpl 109	. . . . 5 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = ⟨𝑥, 𝑦⟩)
4	3	eleq2d 2301	. . . 4 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (∅ ∈ 𝐶 ↔ ∅ ∈ ⟨𝑥, 𝑦⟩))
5	2, 4	mtbiri 681	. . 3 ⊢ ((𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
6	5	exlimivv 1945	. 2 ⊢ (∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶)
7	1, 6	sylbi 121	1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494  ⟨cop 3672   × cxp 4723

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731

This theorem is referenced by:  dmsn0el  5206