Theorem rabtru 28723

Description: Abtract builder using the constant wff ⊤ (Contributed by Thierry Arnoux, 4-May-2020.)

Hypothesis

Ref	Expression
rabtru.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
rabtru	⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Proof of Theorem rabtru

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝑦
2		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nftru 1721	. . . 4 ⊢ Ⅎ𝑥⊤
4		biidd 251	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
5	1, 2, 3, 4	elrabf 3329	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
6		tru 1479	. . . 4 ⊢ ⊤
7	6	biantru 525	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
8	5, 7	bitr4i 266	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
9	8	eqriv 2607	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Ⅎwnfc 2738  {crab 2900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175

This theorem is referenced by:  mptexgf  28809  aciunf1  28845