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Theorem rabsssn 4564
 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 3079 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4523 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2sseq12i 3922 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋})
4 ss2ab 3964 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋))
5 impexp 454 . . . 4 (((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ (𝑥𝑉 → (𝜑𝑥 = 𝑋)))
65albii 1821 . . 3 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
7 df-ral 3075 . . 3 (∀𝑥𝑉 (𝜑𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
86, 7bitr4i 281 . 2 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
93, 4, 83bitri 300 1 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  {crab 3074   ⊆ wss 3858  {csn 4522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-in 3865  df-ss 3875  df-sn 4523 This theorem is referenced by:  suppmptcfin  45170  linc1  45221
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