Theorem bj-xpima1sn 37011

Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 6140 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-xpima1sn	⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Proof of Theorem bj-xpima1sn

Step	Hyp	Ref	Expression
1		bj-xpimasn 37010	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		iffalse 4485	. 2 ⊢ (¬ 𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = ∅)
3	1, 2	eqtrid 2780	1 ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  ∅c0 4284  ifcif 4476  {csn 4577   × cxp 5619   “ cima 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634

This theorem is referenced by:  bj-projval  37051