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Theorem rabsssn 4186
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 2916 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4149 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2sseq12i 3610 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋})
4 ss2ab 3649 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋))
5 impexp 462 . . . 4 (((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ (𝑥𝑉 → (𝜑𝑥 = 𝑋)))
65albii 1744 . . 3 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
7 df-ral 2912 . . 3 (∀𝑥𝑉 (𝜑𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
86, 7bitr4i 267 . 2 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
93, 4, 83bitri 286 1 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ ∀𝑥𝑉 (𝜑𝑥 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  {crab 2911  wss 3555  {csn 4148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-in 3562  df-ss 3569  df-sn 4149
This theorem is referenced by:  suppmptcfin  41448  linc1  41502
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