Theorem bj-rabtr 33829

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

Assertion

Ref	Expression
bj-rabtr	⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtr

Step	Hyp	Ref	Expression
1		ssrab2 3981	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 3914	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1526	. . . 4 ⊢ ⊤
4	3	rgenw 3117	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 3974	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 707	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3909	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1522  ⊤wtru 1523  ∀wral 3105  {crab 3109   ⊆ wss 3863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-in 3870  df-ss 3878

This theorem is referenced by: (None)