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Theorem ssex 3941
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 3922 (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 2997 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ssex.1 . . . 4 𝐵 ∈ V
32inex2 3939 . . 3 (𝐴𝐵) ∈ V
4 eleq1 2145 . . 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 1285  wcel 1434  Vcvv 2612  cin 2983  wss 2984
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997
This theorem is referenced by:  ssexi  3942  ssexg  3943  inteximm  3950  funimaexglem  5049  tfrexlem  6030  elinp  6935  negfi  10483
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