Theorem disjsn2 1837

Description: Intersection of distinct singletons is disjoint.

Assertion

Ref	Expression
disjsn2	⊢ (¬ A = B → ({A} ∩ {B}) = ∅)

Proof of Theorem disjsn2

Step	Hyp	Ref	Expression
1		elsni 1827	. . . 4 ⊢ (B ∈ {A} → B = A)
2	1	cleqcomd 1106	. . 3 ⊢ (B ∈ {A} → A = B)
3	2	con3i 90	. 2 ⊢ (¬ A = B → ¬ B ∈ {A})
4		disjsn 1836	. 2 ⊢ (({A} ∩ {B}) = ∅ ↔ ¬ B ∈ {A})
5	3, 4	sylibr 175	1 ⊢ (¬ A = B → ({A} ∩ {B}) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092   ∩ cin 1486  ∅c0 1707  {csn 1808

This theorem is referenced by:  xpsndisj 2655  phplem2 3404

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812

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