Theorem unixp0 6240

Description: A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)

Assertion

Ref	Expression
unixp0	⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Proof of Theorem unixp0

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

Step	Hyp	Ref	Expression
1		unieq 4881	. . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 4901	. . 3 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2787	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
4		n0 4311	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5		elxp3 5703	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		elssuni 4903	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵))
7		vex 3450	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 3450	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	opnzi 5436	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
10		ssn0 4365	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅)
11	6, 9, 10	sylancl 586	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
12	11	adantl 482	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
13	12	exlimivv 1935	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
14	5, 13	sylbi 216	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
15	14	exlimiv 1933	. . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
16	4, 15	sylbi 216	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅)
17	16	necon4i 2975	. 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
18	3, 17	impbii 208	1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939   ⊆ wss 3913  ∅c0 4287  ⟨cop 4597  ∪ cuni 4870   × cxp 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-opab 5173  df-xp 5644

This theorem is referenced by:  rankxpsuc  9827