Theorem unixp0 5896

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 4649	. . 3 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 4670	. . 3 ⊢ ∪ ∅ = ∅
3	1, 2	syl6eq 2867	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)
4		n0 4143	. . . 4 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
5		elxp3 5382	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		elssuni 4672	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵))
7		vex 3405	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
8		vex 3405	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
9	7, 8	opnzi 5145	. . . . . . . . 9 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅
10		ssn0 4185	. . . . . . . . 9 ⊢ ((⟨𝑥, 𝑦⟩ ⊆ ∪ (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ≠ ∅) → ∪ (𝐴 × 𝐵) ≠ ∅)
11	6, 9, 10	sylancl 576	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
12	11	adantl 469	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
13	12	exlimivv 2023	. . . . . 6 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝑧 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ∪ (𝐴 × 𝐵) ≠ ∅)
14	5, 13	sylbi 208	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
15	14	exlimiv 2021	. . . 4 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∪ (𝐴 × 𝐵) ≠ ∅)
16	4, 15	sylbi 208	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ (𝐴 × 𝐵) ≠ ∅)
17	16	necon4i 3024	. 2 ⊢ (∪ (𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
18	3, 17	impbii 200	1 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   ↔ wb 197   ∧ wa 384   = wceq 1637  ∃wex 1859   ∈ wcel 2157   ≠ wne 2989   ⊆ wss 3780  ∅c0 4127  ⟨cop 4387  ∪ cuni 4641   × cxp 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-opab 4918  df-xp 5330

This theorem is referenced by:  rankxpsuc  9001