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Theorem intmin 3841
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 2727 . . . . 5 𝑦 ∈ V
21elintrab 3833 . . . 4 (𝑦 {𝑥𝐵𝐴𝑥} ↔ ∀𝑥𝐵 (𝐴𝑥𝑦𝑥))
3 ssid 3160 . . . . 5 𝐴𝐴
4 sseq2 3164 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
5 eleq2 2228 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
64, 5imbi12d 233 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥𝑦𝑥) ↔ (𝐴𝐴𝑦𝐴)))
76rspcv 2824 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → (𝐴𝐴𝑦𝐴)))
83, 7mpii 44 . . . 4 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → 𝑦𝐴))
92, 8syl5bi 151 . . 3 (𝐴𝐵 → (𝑦 {𝑥𝐵𝐴𝑥} → 𝑦𝐴))
109ssrdv 3146 . 2 (𝐴𝐵 {𝑥𝐵𝐴𝑥} ⊆ 𝐴)
11 ssintub 3839 . . 3 𝐴 {𝑥𝐵𝐴𝑥}
1211a1i 9 . 2 (𝐴𝐵𝐴 {𝑥𝐵𝐴𝑥})
1310, 12eqssd 3157 1 (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1342  wcel 2135  wral 2442  {crab 2446  wss 3114   cint 3821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rab 2451  df-v 2726  df-in 3120  df-ss 3127  df-int 3822
This theorem is referenced by:  intmin2  3847  bm2.5ii  4470  onsucmin  4481  cldcls  12712
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