Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-nuliotaALT Structured version   Unicode version

Theorem bj-nuliotaALT 34988
Description: Alternate proof of bj-nuliota 34987. Note that this alternate proof uses the fact that  iota x ph evaluates to  (/) when there is no  x satisfying  ph (iotanul 5549). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-nuliotaALT  |-  (/)  =  ( iota x A. y  -.  y  e.  x
)
Distinct variable group:    x, y

Proof of Theorem bj-nuliotaALT
StepHypRef Expression
1 0ss 3813 . 2  |-  (/)  C_  ( iota x A. y  -.  y  e.  x )
2 iotassuni 5550 . . 3  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  U. { x  | 
A. y  -.  y  e.  x }
3 eq0 3799 . . . . . . 7  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
43bicomi 202 . . . . . 6  |-  ( A. y  -.  y  e.  x  <->  x  =  (/) )
54abbii 2588 . . . . 5  |-  { x  |  A. y  -.  y  e.  x }  =  {
x  |  x  =  (/) }
65unieqi 4244 . . . 4  |-  U. {
x  |  A. y  -.  y  e.  x }  =  U. { x  |  x  =  (/) }
7 df-sn 4017 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
87eqcomi 2467 . . . . 5  |-  { x  |  x  =  (/) }  =  { (/) }
98unieqi 4244 . . . 4  |-  U. {
x  |  x  =  (/) }  =  U. { (/)
}
10 0ex 4569 . . . . 5  |-  (/)  e.  _V
1110unisn 4250 . . . 4  |-  U. { (/)
}  =  (/)
126, 9, 113eqtri 2487 . . 3  |-  U. {
x  |  A. y  -.  y  e.  x }  =  (/)
132, 12sseqtri 3521 . 2  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  (/)
141, 13eqssi 3505 1  |-  (/)  =  ( iota x A. y  -.  y  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1396    = wceq 1398   {cab 2439   (/)c0 3783   {csn 4016   U.cuni 4235   iotacio 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-uni 4236  df-iota 5534
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator