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Theorem intmin 3609
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 2538 . . . . 5 y V
21elintrab 3601 . . . 4 (y {x BAx} ↔ x B (Axy x))
3 ssid 2941 . . . . 5 AA
4 sseq2 2944 . . . . . . 7 (x = A → (AxAA))
5 eleq2 2083 . . . . . . 7 (x = A → (y xy A))
64, 5imbi12d 223 . . . . . 6 (x = A → ((Axy x) ↔ (AAy A)))
76rspcv 2629 . . . . 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 2928 . 2 (A B {x BAx} ⊆ A)
11 ssintub 3607 . . 3 A {x BAx}
1211a1i 9 . 2 (A BA {x BAx})
1310, 12eqssd 2939 1 (A B {x BAx} = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  wral 2284  {crab 2288  wss 2894   cint 3589
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-in 2901  df-ss 2908  df-int 3590
This theorem is referenced by:  intmin2  3615  bm2.5ii  4172  onsucmin  4182
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