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Theorem difrab0eqim 3282
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 = {x 𝑉φ} → (𝑉 ∖ {x 𝑉φ}) = ∅)
Distinct variable group:   x,𝑉
Allowed substitution hint:   φ(x)

Proof of Theorem difrab0eqim
StepHypRef Expression
1 ssrabeq 3020 . 2 (𝑉 ⊆ {x 𝑉φ} ↔ 𝑉 = {x 𝑉φ})
2 ssdif0im 3280 . 2 (𝑉 ⊆ {x 𝑉φ} → (𝑉 ∖ {x 𝑉φ}) = ∅)
31, 2sylbir 125 1 (𝑉 = {x 𝑉φ} → (𝑉 ∖ {x 𝑉φ}) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {crab 2304  cdif 2908  wss 2911  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by: (None)
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