Theorem 0xp 4740

Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
0xp	⊢ (∅ × 𝐴) = ∅

Proof of Theorem 0xp

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

Step	Hyp	Ref	Expression
1		elxp 4677	. . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)))
2		noel 3451	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
3		simprl 529	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅)
4	2, 3	mto 663	. . . . . 6 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
5	4	nex 1511	. . . . 5 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
6	5	nex 1511	. . . 4 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))
7		noel 3451	. . . 4 ⊢ ¬ 𝑧 ∈ ∅
8	6, 7	2false 702	. . 3 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) ↔ 𝑧 ∈ ∅)
9	1, 8	bitri 184	. 2 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅)
10	9	eqriv 2190	1 ⊢ (∅ × 𝐴) = ∅

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∅c0 3447  ⟨cop 3622   × cxp 4658

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-xp 4666

This theorem is referenced by:  res0  4947  xp0  5086  xpeq0r  5089  xpdisj1  5091  xpima1  5113  xpfi  6988  exmidfodomrlemim  7263  hashxp  10900