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Theorem ssrabeq 3020
Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
ssrabeq (𝑉 ⊆ {x 𝑉φ} ↔ 𝑉 = {x 𝑉φ})
Distinct variable group:   x,𝑉
Allowed substitution hint:   φ(x)

Proof of Theorem ssrabeq
StepHypRef Expression
1 ssrab2 3019 . . 3 {x 𝑉φ} ⊆ 𝑉
21biantru 286 . 2 (𝑉 ⊆ {x 𝑉φ} ↔ (𝑉 ⊆ {x 𝑉φ} {x 𝑉φ} ⊆ 𝑉))
3 eqss 2954 . 2 (𝑉 = {x 𝑉φ} ↔ (𝑉 ⊆ {x 𝑉φ} {x 𝑉φ} ⊆ 𝑉))
42, 3bitr4i 176 1 (𝑉 ⊆ {x 𝑉φ} ↔ 𝑉 = {x 𝑉φ})
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  {crab 2304  wss 2911
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-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-bnd 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-in 2918  df-ss 2925
This theorem is referenced by:  difrab0eqim  3282
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