Theorem bj-2upln0

Description: A couple is nonempty. (Contributed by BJ, 21-Apr-2019)

	Ref	Expression
Assertion	bj-2upln0	⊢ ⦅ 𝐴 , 𝐵 ⦆ ≠ ∅

Proof

Step	Hyp	Ref	Expression
1		df-bj-2upl	⊢ ⦅ 𝐴 , 𝐵 ⦆ = ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) )
2		bj-1upln0	⊢ ⦅ 𝐴 ⦆ ≠ ∅
3		0pss	⊢ ( ∅ ⊊ ⦅ 𝐴 ⦆ ↔ ⦅ 𝐴 ⦆ ≠ ∅ )
4	2 3	mpbir	⊢ ∅ ⊊ ⦅ 𝐴 ⦆
5		ssun1	⊢ ⦅ 𝐴 ⦆ ⊆ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) )
6		psssstr	⊢ ( ( ∅ ⊊ ⦅ 𝐴 ⦆ ∧ ⦅ 𝐴 ⦆ ⊆ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) ) ) → ∅ ⊊ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) ) )
7	4 5 6	mp2an	⊢ ∅ ⊊ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) )
8		0pss	⊢ ( ∅ ⊊ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) ) ↔ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) ) ≠ ∅ )
9	7 8	mpbi	⊢ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐵 ) ) ≠ ∅
10	1 9	eqnetri	⊢ ⦅ 𝐴 , 𝐵 ⦆ ≠ ∅