Theorem bj-xpima1snALT 37454

Description: Alternate proof of bj-xpima1sn 37453. (Contributed by BJ, 6-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-xpima1snALT	⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Proof of Theorem bj-xpima1snALT

Step	Hyp	Ref	Expression
1		disjsn 4673	. 2 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
2		xpima1 6173	. 2 ⊢ ((𝐴 ∩ {𝑋}) = ∅ → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
3	1, 2	sylbir 238	1 ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1563   ∈ wcel 2145   ∩ cin 3906  ∅c0 4288  {csn 4585   × cxp 5650   “ cima 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by: (None)