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Theorem bj-xpima1snALT 33802
 Description: Alternate proof of bj-xpima1sn 33801. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-xpima1snALT 𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Proof of Theorem bj-xpima1snALT
StepHypRef Expression
1 disjsn 4517 . 2 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
2 xpima1 5877 . 2 ((𝐴 ∩ {𝑋}) = ∅ → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
31, 2sylbir 227 1 𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1508   ∈ wcel 2051   ∩ cin 3821  ∅c0 4172  {csn 4435   × cxp 5401   “ cima 5406 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416 This theorem is referenced by: (None)
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