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Theorem intmin2 4975
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 3512 . . 3 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
21inteqi 4950 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
3 intmin2.1 . . 3 𝐴 ∈ V
4 intmin 4968 . . 3 (𝐴 ∈ V → {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴)
53, 4ax-mp 5 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴
62, 5eqtr3i 2767 1 {𝑥𝐴𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  {crab 3436  Vcvv 3480  wss 3951   cint 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-int 4947
This theorem is referenced by:  dfid7  43625
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