Theorem bnj521

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	bnj521	⊢ ( 𝐴 ∩ { 𝐴 } ) = ∅

Proof

Step	Hyp	Ref	Expression
1		elirr	⊢ ¬ 𝐴 ∈ 𝐴
2		elin	⊢ ( 𝑥 ∈ ( 𝐴 ∩ { 𝐴 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ { 𝐴 } ) )
3		velsn	⊢ ( 𝑥 ∈ { 𝐴 } ↔ 𝑥 = 𝐴 )
4		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴 ) )
5	4	biimpac	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐴 ) → 𝐴 ∈ 𝐴 )
6	3 5	sylan2b	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ { 𝐴 } ) → 𝐴 ∈ 𝐴 )
7	2 6	sylbi	⊢ ( 𝑥 ∈ ( 𝐴 ∩ { 𝐴 } ) → 𝐴 ∈ 𝐴 )
8	7	exlimiv	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ { 𝐴 } ) → 𝐴 ∈ 𝐴 )
9	1 8	mto	⊢ ¬ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ { 𝐴 } )
10		n0	⊢ ( ( 𝐴 ∩ { 𝐴 } ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( 𝐴 ∩ { 𝐴 } ) )
11	9 10	mtbir	⊢ ¬ ( 𝐴 ∩ { 𝐴 } ) ≠ ∅
12		nne	⊢ ( ¬ ( 𝐴 ∩ { 𝐴 } ) ≠ ∅ ↔ ( 𝐴 ∩ { 𝐴 } ) = ∅ )
13	11 12	mpbi	⊢ ( 𝐴 ∩ { 𝐴 } ) = ∅