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Theorem intmin2 4980
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 3510 . . 3 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
21inteqi 4955 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
3 intmin2.1 . . 3 𝐴 ∈ V
4 intmin 4973 . . 3 (𝐴 ∈ V → {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴)
53, 4ax-mp 5 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴
62, 5eqtr3i 2765 1 {𝑥𝐴𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  {crab 3433  Vcvv 3478  wss 3963   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-ss 3980  df-int 4952
This theorem is referenced by:  dfid7  43602
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