Theorem bj-rabtrALT 37624

Description: Alternate proof of bj-rabtr 37623. (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 3438	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤}
2		nfcv 2927	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 2955	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴))
4		tru 1574	. . 3 ⊢ ⊤
5		rabid 3439	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	4, 5	mpbiran2 723	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)
7	3, 6	mpgbir 1832	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  {crab 3418

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419

This theorem is used by: (None)