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Theorem basis1 17007
 Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)
Assertion
Ref Expression
basis1

Proof of Theorem basis1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisg 17004 . . . 4
21ibi 233 . . 3
3 ineq1 3527 . . . . 5
43pweqd 3796 . . . . . . 7
54ineq2d 3534 . . . . . 6
65unieqd 4018 . . . . 5
73, 6sseq12d 3369 . . . 4
8 ineq2 3528 . . . . 5
98pweqd 3796 . . . . . . 7
109ineq2d 3534 . . . . . 6
1110unieqd 4018 . . . . 5
128, 11sseq12d 3369 . . . 4
137, 12rspc2v 3050 . . 3
142, 13syl5com 28 . 2
15143impib 1151 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697   cin 3311   wss 3312  cpw 3791  cuni 4007  ctb 16954 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-bases 16957
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