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Theorem rabsssn 4633
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 3411 . . 3 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
2 df-sn 4592 . . 3 {𝑋} = {𝑥𝑥 = 𝑋}
31, 2sseq12i 3979 . 2 ({𝑥𝑉𝜑} ⊆ {𝑋} ↔ {𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋})
4 ss2ab 4021 . 2 ({𝑥 ∣ (𝑥𝑉𝜑)} ⊆ {𝑥𝑥 = 𝑋} ↔ ∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋))
5 impexp 452 . . . 4 (((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ (𝑥𝑉 → (𝜑𝑥 = 𝑋)))
65albii 1822 . . 3 (∀𝑥((𝑥𝑉𝜑) → 𝑥 = 𝑋) ↔ ∀𝑥(𝑥𝑉 → (𝜑𝑥 = 𝑋)))
7 df-ral 3066 . . 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 205  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2714  wral 3065  {crab 3410  wss 3915  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-sn 4592
This theorem is referenced by:  suppmptcfin  46529  linc1  46580
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