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Theorem intmin2 3833
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 2733 . . 3 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
21inteqi 3811 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
3 intmin2.1 . . 3 𝐴 ∈ V
4 intmin 3827 . . 3 (𝐴 ∈ V → {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴)
53, 4ax-mp 5 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴
62, 5eqtr3i 2180 1 {𝑥𝐴𝑥} = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  {cab 2143  {crab 2439  Vcvv 2712  wss 3102   cint 3807
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-in 3108  df-ss 3115  df-int 3808
This theorem is referenced by: (None)
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