Theorem rabtru 3679

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		tru 1543	. . 3 ⊢ ⊤
2		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥𝑦
3		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nftru 1804	. . . 4 ⊢ Ⅎ𝑥⊤
5		biidd 261	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
6	2, 3, 4, 5	elrabf 3678	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
7	1, 6	mpbiran2 706	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
8	7	eqriv 2727	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1539  ⊤wtru 1540   ∈ wcel 2104  Ⅎwnfc 2881  {crab 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474

This theorem is referenced by:  mptexgf  7225  aciunf1  32155