Theorem rabtru 3590

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 1546	. . 3 ⊢ ⊤
2		nfcv 2900	. . . 4 ⊢ Ⅎ𝑥𝑦
3		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nftru 1811	. . . 4 ⊢ Ⅎ𝑥⊤
5		biidd 265	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
6	2, 3, 4, 5	elrabf 3589	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
7	1, 6	mpbiran2 710	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
8	7	eqriv 2736	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

Syntax hints:   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  Ⅎwnfc 2880  {crab 3058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-rab 3063  df-v 3402

This theorem is referenced by:  mptexgf  7008  aciunf1  30588