Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabtru Structured version   Visualization version   GIF version

Theorem rabtru 3499
 Description: Abstract builder using the constant wff ⊤ (Contributed by Thierry Arnoux, 4-May-2020.)
Hypothesis
Ref Expression
rabtru.1 𝑥𝐴
Assertion
Ref Expression
rabtru {𝑥𝐴 ∣ ⊤} = 𝐴

Proof of Theorem rabtru
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2900 . . . 4 𝑥𝑦
2 rabtru.1 . . . 4 𝑥𝐴
3 nftru 1877 . . . 4 𝑥
4 biidd 252 . . . 4 (𝑥 = 𝑦 → (⊤ ↔ ⊤))
51, 2, 3, 4elrabf 3498 . . 3 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ (𝑦𝐴 ∧ ⊤))
6 tru 1634 . . . 4
76biantru 527 . . 3 (𝑦𝐴 ↔ (𝑦𝐴 ∧ ⊤))
85, 7bitr4i 267 . 2 (𝑦 ∈ {𝑥𝐴 ∣ ⊤} ↔ 𝑦𝐴)
98eqriv 2755 1 {𝑥𝐴 ∣ ⊤} = 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1630  ⊤wtru 1631   ∈ wcel 2137  Ⅎwnfc 2887  {crab 3052 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-rab 3057  df-v 3340 This theorem is referenced by:  mptexgf  6647  aciunf1  29770
 Copyright terms: Public domain W3C validator