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Theorem ssiinf 4141
 Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
Hypothesis
Ref Expression
ssiinf.1
Assertion
Ref Expression
ssiinf

Proof of Theorem ssiinf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2960 . . . . 5
2 eliin 4099 . . . . 5
31, 2ax-mp 8 . . . 4
43ralbii 2730 . . 3
5 ssiinf.1 . . . 4
6 nfcv 2573 . . . 4
75, 6ralcomf 2867 . . 3
84, 7bitri 242 . 2
9 dfss3 3339 . 2
10 dfss3 3339 . . 3
1110ralbii 2730 . 2
128, 9, 113bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wb 178   wcel 1726  wnfc 2560  wral 2706  cvv 2957   wss 3321  ciin 4095 This theorem is referenced by:  ssiin  4142  dmiin  5114 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-in 3328  df-ss 3335  df-iin 4097
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