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

Theorem rabsssn 4673
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 3434 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4632 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2sseq12i 4026 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋})
4 ss2ab 4072 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋))
5 impexp 450 . . . 4 (((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ (𝑥𝑉 → (𝜑𝑥 = 𝑋)))
65albii 1816 . . 3 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
7 df-ral 3060 . . 3 (∀𝑥𝑉 (𝜑𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
86, 7bitr4i 278 . 2 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
93, 4, 83bitri 297 1 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  {crab 3433  wss 3963  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-ss 3980  df-sn 4632
This theorem is referenced by:  constrfin  33751  suppmptcfin  48221  linc1  48271
  Copyright terms: Public domain W3C validator