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Theorem intmin 3693
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 2618 . . . . 5 𝑦 ∈ V
21elintrab 3685 . . . 4 (𝑦 {𝑥𝐵𝐴𝑥} ↔ ∀𝑥𝐵 (𝐴𝑥𝑦𝑥))
3 ssid 3033 . . . . 5 𝐴𝐴
4 sseq2 3037 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥𝐴𝐴))
5 eleq2 2148 . . . . . . 7 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
64, 5imbi12d 232 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥𝑦𝑥) ↔ (𝐴𝐴𝑦𝐴)))
76rspcv 2711 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → (𝐴𝐴𝑦𝐴)))
83, 7mpii 43 . . . 4 (𝐴𝐵 → (∀𝑥𝐵 (𝐴𝑥𝑦𝑥) → 𝑦𝐴))
92, 8syl5bi 150 . . 3 (𝐴𝐵 → (𝑦 {𝑥𝐵𝐴𝑥} → 𝑦𝐴))
109ssrdv 3020 . 2 (𝐴𝐵 {𝑥𝐵𝐴𝑥} ⊆ 𝐴)
11 ssintub 3691 . . 3 𝐴 {𝑥𝐵𝐴𝑥}
1211a1i 9 . 2 (𝐴𝐵𝐴 {𝑥𝐵𝐴𝑥})
1310, 12eqssd 3031 1 (𝐴𝐵 {𝑥𝐵𝐴𝑥} = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  wral 2355  {crab 2359  wss 2988   cint 3673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rab 2364  df-v 2617  df-in 2994  df-ss 3001  df-int 3674
This theorem is referenced by:  intmin2  3699  bm2.5ii  4288  onsucmin  4299
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