Theorem bj-rabtr 37682

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 4031	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 3956	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		tru 1574	. . . 4 ⊢ ⊤
4	3	rgenw 3082	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ⊤
5		ssrab 4022	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⊤))
6	2, 4, 5	mpbir2an 724	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 3950	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571  ∀wral 3078  {crab 3414   ⊆ wss 3902

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rab 3415  df-ss 3919

This theorem is used by: (None)