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Theorem intmin2 4875
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 3499 . . 3 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
21inteqi 4852 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = {𝑥𝐴𝑥}
3 intmin2.1 . . 3 𝐴 ∈ V
4 intmin 4868 . . 3 (𝐴 ∈ V → {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴)
53, 4ax-mp 5 . 2 {𝑥 ∈ V ∣ 𝐴𝑥} = 𝐴
62, 5eqtr3i 2845 1 {𝑥𝐴𝑥} = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2114  {cab 2798  {crab 3129  Vcvv 3470  wss 3909   cint 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3472  df-in 3916  df-ss 3926  df-int 4849
This theorem is referenced by:  dfid7  40103
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