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Theorem bnd2 3917
Description: A variant of the Boundedness Axiom bnd 3916 that picks a subset out of a possibly proper class in which a property is true. (Contributed by NM, 4-Feb-2004.)
Hypothesis
Ref Expression
bnd2.1  _V
Assertion
Ref Expression
bnd2  C_
Distinct variable groups:   ,   ,,   ,,,
Allowed substitution hints:   (,)   ()

Proof of Theorem bnd2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . . . 4
21ralbii 2324 . . 3
3 bnd2.1 . . . 4  _V
4 raleq 2499 . . . . 5
5 raleq 2499 . . . . . 6
65exbidv 1703 . . . . 5
74, 6imbi12d 223 . . . 4
8 bnd 3916 . . . 4
93, 7, 8vtocl 2602 . . 3
102, 9sylbi 114 . 2
11 vex 2554 . . . . 5 
_V
1211inex1 3882 . . . 4  i^i 
_V
13 inss2 3152 . . . . . . 7  i^i  C_
14 sseq1 2960 . . . . . . 7  i^i 
C_  i^i  C_
1513, 14mpbiri 157 . . . . . 6  i^i  C_
1615biantrurd 289 . . . . 5  i^i 
C_
17 rexeq 2500 . . . . . . 7  i^i  i^i
18 elin 3120 . . . . . . . . . 10  i^i
1918anbi1i 431 . . . . . . . . 9  i^i
20 anass 381 . . . . . . . . 9
2119, 20bitri 173 . . . . . . . 8  i^i
2221rexbii2 2329 . . . . . . 7  i^i
2317, 22syl6bb 185 . . . . . 6  i^i
2423ralbidv 2320 . . . . 5  i^i
2516, 24bitr3d 179 . . . 4  i^i  C_
2612, 25spcev 2641 . . 3 
C_
2726exlimiv 1486 . 2  C_
2810, 27syl 14 1  C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301   _Vcvv 2551    i^i cin 2910    C_ wss 2911
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  ax-coll 3863  ax-sep 3866
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-rex 2306  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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