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Theorem bj-xpima1sn 33249
Description: The image of a singleton by a direct product, empty case. [Change and relabel xpimasn 5737 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-xpima1sn (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Proof of Theorem bj-xpima1sn
StepHypRef Expression
1 bj-xpimasn 33248 . 2 ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋𝐴, 𝐵, ∅)
2 df-nel 3036 . . 3 (𝑋𝐴 ↔ ¬ 𝑋𝐴)
3 iffalse 4239 . . 3 𝑋𝐴 → if(𝑋𝐴, 𝐵, ∅) = ∅)
42, 3sylbi 207 . 2 (𝑋𝐴 → if(𝑋𝐴, 𝐵, ∅) = ∅)
51, 4syl5eq 2806 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1632  wcel 2139  wnel 3035  c0 4058  ifcif 4230  {csn 4321   × cxp 5264  cima 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279
This theorem is referenced by:  bj-projval  33290
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