Theorem bj-xpima1sn 32136

Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5498 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 32135	. 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅)
2		df-nel 2783	. . 3 ⊢ (𝑋 ∉ 𝐴 ↔ ¬ 𝑋 ∈ 𝐴)
3		iffalse 4045	. . 3 ⊢ (¬ 𝑋 ∈ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = ∅)
4	2, 3	sylbi 206	. 2 ⊢ (𝑋 ∉ 𝐴 → if(𝑋 ∈ 𝐴, 𝐵, ∅) = ∅)
5	1, 4	syl5eq 2656	1 ⊢ (𝑋 ∉ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∅c0 3874  ifcif 4036  {csn 4125   × cxp 5036   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by:  bj-projval  32177