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Theorem intmin 4862
 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 3445 . . . . 5 𝑦 ∈ V
21elintrab 4854 . . . 4 (𝑦 {𝑥𝐵𝐴𝑥} ↔ ∀𝑥𝐵 (𝐴𝑥𝑦𝑥))
3 ssid 3939 . . . . 5 𝐴𝐴
4 sseq2 3943 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
5 eleq2 2878 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
64, 5imbi12d 348 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥𝑦𝑥) ↔ (𝐴𝐴𝑦𝐴)))
76rspcv 3567 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → (𝐴𝐴𝑦𝐴)))
83, 7mpii 46 . . . 4 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → 𝑦𝐴))
92, 8syl5bi 245 . . 3 (𝐴𝐵 → (𝑦 {𝑥𝐵𝐴𝑥} → 𝑦𝐴))
109ssrdv 3923 . 2 (𝐴𝐵 {𝑥𝐵𝐴𝑥} ⊆ 𝐴)
11 ssintub 4860 . . 3 𝐴 {𝑥𝐵𝐴𝑥}
1211a1i 11 . 2 (𝐴𝐵𝐴 {𝑥𝐵𝐴𝑥})
1310, 12eqssd 3934 1 (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110   ⊆ wss 3883  ∩ cint 4842 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3444  df-in 3890  df-ss 3900  df-int 4843 This theorem is referenced by:  intmin2  4869  ordintdif  6215  uniordint  7514  onsucmin  7529  rankonidlem  9259  rankval4  9298  harsucnn  9429  mrcid  16896  lspid  19768  aspid  20583  cldcls  21688  spanid  29174  chsupid  29239  igenidl2  35654  pclidN  37343  diaocN  38572
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