MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dtruALT2 Unicode version

Theorem dtruALT2 4219
Description: An alternative proof of dtru 4201 ("two things exist") using ax-pr 4214 instead of ax-pow 4188. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT2  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruALT2
StepHypRef Expression
1 0inp0 4182 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 snex 4216 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2292 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 285 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 2875 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 15 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4150 . . . 4  |-  (/)  e.  _V
8 eqeq2 2292 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 285 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 2875 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 156 . 2  |-  E. x  -.  y  =  x
12 exnal 1561 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2285 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1553 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 301 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 199 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1527   E.wex 1528    = wceq 1623   (/)c0 3455   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647
  Copyright terms: Public domain W3C validator