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

Theorem rabsssn 4162
 Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)
Assertion
Ref Expression
rabsssn ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
Distinct variable group:   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rabsssn
StepHypRef Expression
1 df-rab 2905 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4126 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2sseq12i 3594 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋})
4 ss2ab 3633 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋))
5 impexp 461 . . . 4 (((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ (𝑥𝑉 → (𝜑𝑥 = 𝑋)))
65albii 1737 . . 3 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
7 df-ral 2901 . . 3 (∀𝑥𝑉 (𝜑𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
86, 7bitr4i 266 . 2 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
93, 4, 83bitri 285 1 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  {crab 2900   ⊆ wss 3540  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-in 3547  df-ss 3554  df-sn 4126 This theorem is referenced by:  suppmptcfin  41954  linc1  42008
 Copyright terms: Public domain W3C validator