Theorem bj-rabtr 37427

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 4036	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 3961	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1567	. . . 4 ⊢ ⊤
4	3	rgenw 3083	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 4027	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 723	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3955	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1563  ⊤wtru 1564  ∀wral 3079  {crab 3417   ⊆ wss 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rab 3418  df-ss 3924

This theorem is referenced by: (None)