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Theorem bj-nuliotaALT 31624
Description: Alternate proof of bj-nuliota 31623. Note that this alternate proof uses the fact that  iota x ph evaluates to  (/) when there is no  x satisfying  ph (iotanul 5561). 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 3763 . 2  |-  (/)  C_  ( iota x A. y  -.  y  e.  x )
2 iotassuni 5562 . . 3  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  U. { x  | 
A. y  -.  y  e.  x }
3 eq0 3747 . . . . . . 7  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
43bicomi 206 . . . . . 6  |-  ( A. y  -.  y  e.  x  <->  x  =  (/) )
54abbii 2567 . . . . 5  |-  { x  |  A. y  -.  y  e.  x }  =  {
x  |  x  =  (/) }
65unieqi 4207 . . . 4  |-  U. {
x  |  A. y  -.  y  e.  x }  =  U. { x  |  x  =  (/) }
7 df-sn 3969 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
87eqcomi 2460 . . . . 5  |-  { x  |  x  =  (/) }  =  { (/) }
98unieqi 4207 . . . 4  |-  U. {
x  |  x  =  (/) }  =  U. { (/)
}
10 0ex 4535 . . . . 5  |-  (/)  e.  _V
1110unisn 4213 . . . 4  |-  U. { (/)
}  =  (/)
126, 9, 113eqtri 2477 . . 3  |-  U. {
x  |  A. y  -.  y  e.  x }  =  (/)
132, 12sseqtri 3464 . 2  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  (/)
141, 13eqssi 3448 1  |-  (/)  =  ( iota x A. y  -.  y  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1442    = wceq 1444   {cab 2437   (/)c0 3731   {csn 3968   U.cuni 4198   iotacio 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-nul 4534
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969  df-pr 3971  df-uni 4199  df-iota 5546
This theorem is referenced by: (None)
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