Theorem bj-rabtr 32926

Description: Restricted class abstraction with true formula. (Contributed by BJ, 22-Apr-2019.)

Assertion

Ref	Expression
bj-rabtr	|- { x e. A | T. } = A

Distinct variable group:   x, A

Proof of Theorem bj-rabtr

Step	Hyp	Ref	Expression
1		ssrab2 3687	. 2 |- { x e. A | T. } C_ A
2		ssid 3624	. . 3 |- A C_ A
3		tru 1487	. . . 4 |- T.
4	3	rgenw 2924	. . 3 |- A. x e. A T.
5		ssrab 3680	. . 3 |- ( A C_ { x e. A | T. } <-> ( A C_ A /\ A. x e. A T. ) )
6	2, 4, 5	mpbir2an 955	. 2 |- A C_ { x e. A | T. }
7	1, 6	eqssi 3619	1 |- { x e. A | T. } = A

Syntax hints:   = wceq 1483   T. wtru 1484   A.wral 2912   {crab 2916   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by: (None)