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Theorem ssex 4072
 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 4053 (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 3088 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ssex.1 . . . 4 𝐵 ∈ V
32inex2 4070 . . 3 (𝐴𝐵) ∈ V
4 eleq1 2203 . . 3 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ∈ V ↔ 𝐴 ∈ V))
53, 4mpbii 147 . 2 ((𝐴𝐵) = 𝐴𝐴 ∈ V)
61, 5sylbi 120 1 (𝐴𝐵𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ∩ cin 3074   ⊆ wss 3075 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081  df-ss 3088 This theorem is referenced by:  ssexi  4073  ssexg  4074  inteximm  4081  funimaexglem  5213  tfrexlem  6238  elinp  7305  suplocexprlem2b  7545  negfi  11030  elcncf  12766  exmid1stab  13366  sbthom  13394
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