Theorem difrab0eqim 3526

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 ∈ 𝑉 ∣ 𝜑} → (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)

Distinct variable group:   𝑥,𝑉

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem difrab0eqim

Step	Hyp	Ref	Expression
1		ssrabeq 3279	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑})
2		ssdif0im 3524	. 2 ⊢ (𝑉 ⊆ {𝑥 ∈ 𝑉 ∣ 𝜑} → (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)
3	1, 2	sylbir 135	1 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ 𝜑} → (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ 𝜑}) = ∅)

Syntax hints:   → wi 4   = wceq 1372  {crab 2487   ∖ cdif 3162   ⊆ wss 3165  ∅c0 3459

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460

This theorem is referenced by: (None)