Theorem bj-rabtrALT 37454

Description: Alternate proof of bj-rabtr 37453. (Contributed by BJ, 22-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-rabtrALT

Step	Hyp	Ref	Expression
1		nfrab1 3443	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤}
2		nfcv 2931	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 2959	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴))
4		tru 1571	. . 3 ⊢ ⊤
5		rabid 3444	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	4, 5	mpbiran2 722	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)
7	3, 6	mpgbir 1826	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ↔ wb 209   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  {crab 3423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424

This theorem is referenced by: (None)