Theorem bj-rabtrALT 34251

Description: Alternate proof of bj-rabtr 34250. (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 3386	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ⊤}
2		nfcv 2979	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	cleqf 3012	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴))
4		tru 1541	. . 3 ⊢ ⊤
5		rabid 3380	. . 3 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
6	4, 5	mpbiran2 708	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑥 ∈ 𝐴)
7	3, 6	mpgbir 1800	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ↔ wb 208   = wceq 1537  ⊤wtru 1538   ∈ wcel 2114  {crab 3144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149

This theorem is referenced by: (None)