Theorem bj-rabtrALT 35806

Description: Alternate proof of bj-rabtr 35805. (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 3451	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤}
2		nfcv 2903	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 2934	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴))
4		tru 1545	. . 3 ⊢ ⊤
5		rabid 3452	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	4, 5	mpbiran2 708	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)
7	3, 6	mpgbir 1801	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ↔ wb 205   = wceq 1541  ⊤wtru 1542   ∈ wcel 2106  {crab 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433

This theorem is referenced by: (None)