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Theorem intmin2 3953
 Description: Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
intmin2.1 A V
Assertion
Ref Expression
intmin2 {x A x} = A
Distinct variable group:   x,A

Proof of Theorem intmin2
StepHypRef Expression
1 rabab 2876 . . 3 {x V A x} = {x A x}
21inteqi 3930 . 2 {x V A x} = {x A x}
3 intmin2.1 . . 3 A V
4 intmin 3946 . . 3 (A V → {x V A x} = A)
53, 4ax-mp 8 . 2 {x V A x} = A
62, 5eqtr3i 2375 1 {x A x} = A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  Vcvv 2859   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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