Theorem rabtru 3466

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 2866	. . . 4 ⊢ Ⅎ𝑥𝑦
2		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nftru 1843	. . . 4 ⊢ Ⅎ𝑥⊤
4		biidd 252	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
5	1, 2, 3, 4	elrabf 3465	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
6		tru 1600	. . . 4 ⊢ ⊤
7	6	biantru 527	. . 3 ⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
8	5, 7	bitr4i 267	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
9	8	eqriv 2721	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   ∧ wa 383   = wceq 1596  ⊤wtru 1597   ∈ wcel 2103  Ⅎwnfc 2853  {crab 3018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rab 3023  df-v 3306

This theorem is referenced by:  mptexgf  6601  aciunf1  29693