Theorem bj-1uplth

Description: The characteristic property of monuples. Note that this holds without sethood hypotheses. (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-1uplth	$$⊢⦅A⦆=⦅B⦆\leftrightarrow A=B$$

Proof

Step	Hyp	Ref	Expression
1		bj-pr1eq	$$⊢⦅A⦆=⦅B⦆\to pr1⦅A⦆=pr1⦅B⦆$$
2		bj-pr11val	$$⊢pr1⦅A⦆=A$$
3		bj-pr11val	$$⊢pr1⦅B⦆=B$$
4	1 2 3	3eqtr3g	$$⊢⦅A⦆=⦅B⦆\to A=B$$
5		bj-1upleq	$$⊢A=B\to ⦅A⦆=⦅B⦆$$
6	4 5	impbii	$$⊢⦅A⦆=⦅B⦆\leftrightarrow A=B$$