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

Theorem dtruALT 4646
Description: Alternate proof of dtru 4608 which requires more axioms but is shorter and may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that  x and  y be distinct. Specifically, theorem spcev 3170 requires that  x must not occur in the subexpression  -.  y  =  { (/) } in step 4 nor in the subexpression  -.  y  =  (/) in step 9. The proof verifier will require that  x and  y be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
dtruALT  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruALT
StepHypRef Expression
1 0inp0 4589 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 p0ex 4604 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2435 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 295 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 3170 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 17 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4549 . . . 4  |-  (/)  e.  _V
8 eqeq2 2435 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 295 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 3170 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 167 . 2  |-  E. x  -.  y  =  x
12 exnal 1695 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2429 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1687 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 311 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 211 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1435    = wceq 1437   E.wex 1659   (/)c0 3758   {csn 3993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-v 3080  df-dif 3436  df-in 3440  df-ss 3447  df-nul 3759  df-pw 3978  df-sn 3994
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator