Theorem bj-rabtr 38034

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 4008	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 3935	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1574	. . . 4 ⊢ ⊤
4	3	rgenw 3084	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 3999	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 724	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3929	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571  ∀wral 3080  {crab 3416   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rab 3417  df-ss 3901

This theorem is used by: (None)