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Theorem bj-xpima1snALT 32618
Description: Alternate proof of bj-xpima1sn 32617. (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 4221 . . 3 ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋𝐴)
2 df-nel 2894 . . 3 (𝑋𝐴 ↔ ¬ 𝑋𝐴)
31, 2bitr4i 267 . 2 ((𝐴 ∩ {𝑋}) = ∅ ↔ 𝑋𝐴)
4 xpima1 5541 . 2 ((𝐴 ∩ {𝑋}) = ∅ → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
53, 4sylbir 225 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  wnel 2893  cin 3558  c0 3896  {csn 4153   × cxp 5077  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by: (None)
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