Theorem bj-rabtr 36114

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 4077	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 4004	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1544	. . . 4 ⊢ ⊤
4	3	rgenw 3064	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 4070	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 708	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3998	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1540  ⊤wtru 1541  ∀wral 3060  {crab 3431   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965

This theorem is referenced by: (None)