Theorem rabtru 3361

Description: Abstract 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 2764	. . . 4 ⊢ Ⅎ𝑥𝑦
2		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nftru 1730	. . . 4 ⊢ Ⅎ𝑥⊤
4		biidd 252	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
5	1, 2, 3, 4	elrabf 3360	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
6		tru 1487	. . . 4 ⊢ ⊤
7	6	biantru 526	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
8	5, 7	bitr4i 267	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
9	8	eqriv 2619	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Ⅎwnfc 2751  {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202

This theorem is referenced by:  mptexgf  6485  aciunf1  29463