Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  axpweq Unicode version

Theorem axpweq 4095
 Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4098 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
Hypothesis
Ref Expression
axpweq.1
Assertion
Ref Expression
axpweq
Distinct variable group:   ,,,

Proof of Theorem axpweq
StepHypRef Expression
1 pwidg 3524 . . . 4
2 pweq 3513 . . . . . 6
32eleq2d 2209 . . . . 5
43spcegv 2774 . . . 4
51, 4mpd 13 . . 3
6 elex 2697 . . . 4
76exlimiv 1577 . . 3
85, 7impbii 125 . 2
9 vex 2689 . . . . 5
109elpw2 4082 . . . 4
11 pwss 3526 . . . . 5
12 dfss2 3086 . . . . . . 7
1312imbi1i 237 . . . . . 6
1413albii 1446 . . . . 5
1511, 14bitri 183 . . . 4
1610, 15bitri 183 . . 3
1716exbii 1584 . 2
188, 17bitri 183 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104  wal 1329   wceq 1331  wex 1468   wcel 1480  cvv 2686   wss 3071  cpw 3510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator