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Theorem intmin 3635
 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
Distinct variable groups:   ,   ,

Proof of Theorem intmin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . 5
21elintrab 3627 . . . 4
3 ssid 2964 . . . . 5
4 sseq2 2967 . . . . . . 7
5 eleq2 2101 . . . . . . 7
64, 5imbi12d 223 . . . . . 6
76rspcv 2652 . . . . 5
83, 7mpii 39 . . . 4
92, 8syl5bi 141 . . 3
109ssrdv 2951 . 2
11 ssintub 3633 . . 3
1211a1i 9 . 2
1310, 12eqssd 2962 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243   wcel 1393  wral 2306  crab 2310   wss 2917  cint 3615 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616 This theorem is referenced by:  intmin2  3641  bm2.5ii  4222  onsucmin  4233
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