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Theorem difrab0eqim 3399
Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
difrab0eqim (𝑉 = {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
Distinct variable group:   𝑥,𝑉
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem difrab0eqim
StepHypRef Expression
1 ssrabeq 3153 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
2 ssdif0im 3397 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
31, 2sylbir 134 1 (𝑉 = {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  {crab 2397  cdif 3038  wss 3041  c0 3333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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