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 1541	. . 3 ⊢ ⊤
2		nfcv 2979	. . . 4 ⊢ Ⅎ𝑥𝑦
3		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nftru 1805	. . . 4 ⊢ Ⅎ𝑥⊤
5		biidd 264	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
6	2, 3, 4, 5	elrabf 3678	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
7	1, 6	mpbiran2 708	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
8	7	eqriv 2820	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1537  ⊤wtru 1538   ∈ wcel 2114  Ⅎwnfc 2963  {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  df-v 3498

This theorem is referenced by:  mptexgf  6987  aciunf1  30410