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Theorem intmin 3626
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 (A B {x BAx} = 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 2554 . . . . 5 y V
21elintrab 3618 . . . 4 (y {x BAx} ↔ x B (Axy x))
3 ssid 2958 . . . . 5 AA
4 sseq2 2961 . . . . . . 7 (x = A → (AxAA))
5 eleq2 2098 . . . . . . 7 (x = A → (y xy A))
64, 5imbi12d 223 . . . . . 6 (x = A → ((Axy x) ↔ (AAy A)))
76rspcv 2646 . . . . 5 (A B → (x B (Axy x) → (AAy A)))
83, 7mpii 39 . . . 4 (A B → (x B (Axy x) → y A))
92, 8syl5bi 141 . . 3 (A B → (y {x BAx} → y A))
109ssrdv 2945 . 2 (A B {x BAx} ⊆ A)
11 ssintub 3624 . . 3 A {x BAx}
1211a1i 9 . 2 (A BA {x BAx})
1310, 12eqssd 2956 1 (A B {x BAx} = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wral 2300  {crab 2304  wss 2911   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by:  intmin2  3632  bm2.5ii  4188  onsucmin  4198
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