Theorem unixp0 6251

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 4876	. . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 4893	. . 3 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2788	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
4		n0 4307	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5		elxp3 5700	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		elssuni 4896	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵))
7		vex 3446	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 3446	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	opnzi 5432	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
10		ssn0 4358	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅)
11	6, 9, 10	sylancl 587	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
12	11	adantl 481	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
13	12	exlimivv 1934	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
14	5, 13	sylbi 217	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
15	14	exlimiv 1932	. . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
16	4, 15	sylbi 217	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅)
17	16	necon4i 2968	. 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
18	3, 17	impbii 209	1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933   ⊆ wss 3903  ∅c0 4287  ⟨cop 4588  ∪ cuni 4865   × cxp 5632

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-opab 5163  df-xp 5640

This theorem is referenced by:  rankxpsuc  9808