Theorem rabtru 3642

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 1574	. . 3 ⊢ ⊤
2		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥𝑦
3		rabtru.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nftru 1837	. . . 4 ⊢ Ⅎ𝑥⊤
5		biidd 265	. . . 4 ⊢ (𝑥 = 𝑦 → (⊤ ↔ ⊤))
6	2, 3, 4, 5	elrabf 3641	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ (𝑦 ∈ 𝐴 ∧ ⊤))
7	1, 6	mpbiran2 723	. 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ 𝑦 ∈ 𝐴)
8	7	eqriv 2757	1 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴

This proof depends on syntax axioms:   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  Ⅎwnfc 2907  {crab 3412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-rab 3413  df-v 3452

This theorem is used by:  mptexgf  7216  aciunf1  33190