Theorem bj-rabtrAUTO 36933

Description: Proof of bj-rabtr 36931 found automatically by the Metamath program "MM-PA> IMPROVE ALL / DEPTH 3 / 3" command followed by "MM-PA> MINIMIZE_WITH *". (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtrAUTO

Step	Hyp	Ref	Expression
1		ssrab2 4080	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} ⊆ 𝐴
2		ssid 4006	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
3	2	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ⊆ 𝐴)
4		simpl 482	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ⊤)
5	3, 4	ssrabdv 4074	. . 3 ⊢ (⊤ → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤})
6	5	mptru 1547	. 2 ⊢ 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ ⊤}
7	1, 6	eqssi 4000	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108  {crab 3436   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3968

This theorem is referenced by: (None)