Theorem bj-xpima1snALT 37626

Description: Alternate proof of bj-xpima1sn 37625. (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 4682	. 2 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴)
2		xpima1 6186	. 2 ⊢ ((𝐴 ∩ {𝑋}) = ∅ → ((𝐴 × 𝐵) “ {𝑋}) = ∅)
3	1, 2	sylbir 238	1 ⊢ (¬ 𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146   ∩ cin 3907  ∅c0 4289  {csn 4594   × cxp 5664   “ cima 5669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679

This theorem is used by: (None)