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Theorem intmin 4972
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 3481 . . . . 5 𝑦 ∈ V
21elintrab 4964 . . . 4 (𝑦 {𝑥𝐵𝐴𝑥} ↔ ∀𝑥𝐵 (𝐴𝑥𝑦𝑥))
3 ssid 4017 . . . . 5 𝐴𝐴
4 sseq2 4021 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
5 eleq2 2827 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
64, 5imbi12d 344 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥𝑦𝑥) ↔ (𝐴𝐴𝑦𝐴)))
76rspcv 3617 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → (𝐴𝐴𝑦𝐴)))
83, 7mpii 46 . . . 4 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → 𝑦𝐴))
92, 8biimtrid 242 . . 3 (𝐴𝐵 → (𝑦 {𝑥𝐵𝐴𝑥} → 𝑦𝐴))
109ssrdv 4000 . 2 (𝐴𝐵 {𝑥𝐵𝐴𝑥} ⊆ 𝐴)
11 ssintub 4970 . . 3 𝐴 {𝑥𝐵𝐴𝑥}
1211a1i 11 . 2 (𝐴𝐵𝐴 {𝑥𝐵𝐴𝑥})
1310, 12eqssd 4012 1 (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wral 3058  {crab 3432  wss 3962   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-ss 3979  df-int 4951
This theorem is referenced by:  intmin2  4979  ordintdif  6435  uniordint  7820  onsucmin  7840  naddrid  8719  naddasslem1  8730  naddasslem2  8731  rankonidlem  9865  rankval4  9904  harsucnn  10035  mrcid  17657  lspid  20997  aspid  21912  cldcls  23065  spanid  31375  chsupid  31440  fldgenidfld  33298  igenidl2  38051  pclidN  39878  diaocN  41107  onuniintrab  43214  topclat  48786  toplatlub  48788  toplatjoin  48790
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