Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  basis1 Structured version   Unicode version

Theorem basis1 19578
 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 19575 . . . 4
21ibi 241 . . 3
3 ineq1 3689 . . . . 5
43pweqd 4020 . . . . . . 7
54ineq2d 3696 . . . . . 6
65unieqd 4261 . . . . 5
73, 6sseq12d 3528 . . . 4
8 ineq2 3690 . . . . 5
98pweqd 4020 . . . . . . 7
109ineq2d 3696 . . . . . 6
1110unieqd 4261 . . . . 5
128, 11sseq12d 3528 . . . 4
137, 12rspc2v 3219 . . 3
142, 13syl5com 30 . 2
15143impib 1194 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1395   wcel 1819  wral 2807   cin 3470   wss 3471  cpw 4015  cuni 4251  ctb 19525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252  df-bases 19528 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator