Theorem unixp0 6271

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 4877	. . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 4895	. . 3 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2814	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
4		n0 4306	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5		elxp3 5714	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		elssuni 4898	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵))
7		vex 3459	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 3459	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	opnzi 5443	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
10		ssn0 4359	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅)
11	6, 9, 10	sylancl 595	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
12	11	adantl 485	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
13	12	exlimivv 1953	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
14	5, 13	sylbi 219	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
15	14	exlimiv 1951	. . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
16	4, 15	sylbi 219	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅)
17	16	necon4i 2993	. 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
18	3, 17	impbii 211	1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143   ≠ wne 2958   ⊆ wss 3905  ∅c0 4286  ⟨cop 4589  ∪ cuni 4866   × cxp 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-opab 5164  df-xp 5654

This theorem is referenced by:  rankxpsuc  9841