Theorem unixp0im 5147

Description: The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
unixp0im	⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)

Proof of Theorem unixp0im

Step	Hyp	Ref	Expression
1		unieq 3805	. 2 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∪ ∅)
2		uni0 3823	. 2 ⊢ ∪ ∅ = ∅
3	1, 2	eqtrdi 2219	1 ⊢ ((𝐴 × 𝐵) = ∅ → ∪ (𝐴 × 𝐵) = ∅)

Syntax hints:   → wi 4   = wceq 1348  ∅c0 3414  ∪ cuni 3796   × cxp 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797

This theorem is referenced by: (None)