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Theorem intmin2 4439
 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 {𝑥𝐴𝑥} = 𝐴
Distinct variable group:   𝑥,𝐴

Proof of Theorem intmin2
StepHypRef Expression
1 rabab 3196 . . 3 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
21inteqi 4414 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
3 intmin2.1 . . 3 𝐴 ∈ V
4 intmin 4432 . . 3 (𝐴 ∈ V → {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴)
53, 4ax-mp 5 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴
62, 5eqtr3i 2634 1 {𝑥𝐴𝑥} = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-int 4411 This theorem is referenced by:  dfid7  36938
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