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Theorem difrab0eqim 3489
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 3242 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})
2 ssdif0im 3487 . 2 (𝑉 ⊆ {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
31, 2sylbir 135 1 (𝑉 = {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  {crab 2459  cdif 3126  wss 3129  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423
This theorem is referenced by: (None)
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