Theorem unixp0 6283

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 4920	. . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 4940	. . 3 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2789	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
4		n0 4347	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5		elxp3 5743	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		elssuni 4942	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵))
7		vex 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	opnzi 5475	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
10		ssn0 4401	. . . . . . . . 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 483	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
13	12	exlimivv 1936	. . . . . 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 1934	. . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
16	4, 15	sylbi 216	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅)
17	16	necon4i 2977	. 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
18	3, 17	impbii 208	1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941   ⊆ wss 3949  ∅c0 4323  ⟨cop 4635  ∪ cuni 4909   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-opab 5212  df-xp 5683

This theorem is referenced by:  rankxpsuc  9877