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Theorem ssex 3951
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 3932 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
Hypothesis
Ref Expression
ssex.1 𝐵 ∈ V
Assertion
Ref Expression
ssex (𝐴𝐵𝐴 ∈ V)

Proof of Theorem ssex
StepHypRef Expression
1 df-ss 3001 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ssex.1 . . . 4 𝐵 ∈ V
32inex2 3949 . . 3 (𝐴𝐵) ∈ V
4 eleq1 2147 . . 3 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ V ↔ 𝐴 ∈ V))
53, 4mpbii 146 . 2 ((𝐴𝐵) = 𝐴𝐴 ∈ V)
61, 5sylbi 119 1 (𝐴𝐵𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  Vcvv 2615  cin 2987  wss 2988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001
This theorem is referenced by:  ssexi  3952  ssexg  3953  inteximm  3960  funimaexglem  5062  tfrexlem  6053  elinp  6977  negfi  10552
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