MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  intmin Structured version   Visualization version   GIF version

Theorem intmin 4927
Description: Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
intmin (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem intmin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3447 . . . . 5 𝑦 ∈ V
21elintrab 4919 . . . 4 (𝑦 {𝑥𝐵𝐴𝑥} ↔ ∀𝑥𝐵 (𝐴𝑥𝑦𝑥))
3 ssid 3964 . . . . 5 𝐴𝐴
4 sseq2 3968 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
5 eleq2 2826 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
64, 5imbi12d 344 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥𝑦𝑥) ↔ (𝐴𝐴𝑦𝐴)))
76rspcv 3575 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → (𝐴𝐴𝑦𝐴)))
83, 7mpii 46 . . . 4 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → 𝑦𝐴))
92, 8biimtrid 241 . . 3 (𝐴𝐵 → (𝑦 {𝑥𝐵𝐴𝑥} → 𝑦𝐴))
109ssrdv 3948 . 2 (𝐴𝐵 {𝑥𝐵𝐴𝑥} ⊆ 𝐴)
11 ssintub 4925 . . 3 𝐴 {𝑥𝐵𝐴𝑥}
1211a1i 11 . 2 (𝐴𝐵𝐴 {𝑥𝐵𝐴𝑥})
1310, 12eqssd 3959 1 (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3062  {crab 3405  wss 3908   cint 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-int 4906
This theorem is referenced by:  intmin2  4934  ordintdif  6365  uniordint  7732  onsucmin  7752  naddid1  8625  naddasslem1  8634  naddasslem2  8635  rankonidlem  9760  rankval4  9799  harsucnn  9930  mrcid  17485  lspid  20428  aspid  21263  cldcls  22377  spanid  30175  chsupid  30240  fldgenidfld  31968  igenidl2  36491  pclidN  38326  diaocN  39555  onuniintrab  41498  topclat  46955  toplatlub  46957  toplatjoin  46959
  Copyright terms: Public domain W3C validator