ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssex GIF version

Theorem ssex 4124
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 4105 (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 3134 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 ssex.1 . . . 4 𝐵 ∈ V
32inex2 4122 . . 3 (𝐴𝐵) ∈ V
4 eleq1 2233 . . 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 1348  wcel 2141  Vcvv 2730  cin 3120  wss 3121
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4105
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134
This theorem is referenced by:  ssexi  4125  ssexg  4126  inteximm  4133  funimaexglem  5279  tfrexlem  6310  elinp  7423  suplocexprlem2b  7663  negfi  11178  ssomct  12387  ssnnctlemct  12388  nninfdc  12395  elcncf  13313  exmid1stab  13993  sbthom  14018
  Copyright terms: Public domain W3C validator