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Theorem bj-nuliotaALT 33543
Description: Alternate proof of bj-nuliota 33542. Note that this alternate proof uses the fact that  iota x ph evaluates to  (/) when there is no  x satisfying  ph (iotanul 5557). 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 3807 . 2  |-  (/)  C_  ( iota x A. y  -.  y  e.  x )
2 iotassuni 5558 . . 3  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  U. { x  | 
A. y  -.  y  e.  x }
3 eq0 3793 . . . . . . 7  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
43bicomi 202 . . . . . 6  |-  ( A. y  -.  y  e.  x  <->  x  =  (/) )
54abbii 2594 . . . . 5  |-  { x  |  A. y  -.  y  e.  x }  =  {
x  |  x  =  (/) }
65unieqi 4247 . . . 4  |-  U. {
x  |  A. y  -.  y  e.  x }  =  U. { x  |  x  =  (/) }
7 df-sn 4021 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
87eqcomi 2473 . . . . 5  |-  { x  |  x  =  (/) }  =  { (/) }
98unieqi 4247 . . . 4  |-  U. {
x  |  x  =  (/) }  =  U. { (/)
}
10 0ex 4570 . . . . 5  |-  (/)  e.  _V
1110unisn 4253 . . . 4  |-  U. { (/)
}  =  (/)
126, 9, 113eqtri 2493 . . 3  |-  U. {
x  |  A. y  -.  y  e.  x }  =  (/)
132, 12sseqtri 3529 . 2  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  (/)
141, 13eqssi 3513 1  |-  (/)  =  ( iota x A. y  -.  y  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1372    = wceq 1374   {cab 2445   (/)c0 3778   {csn 4020   U.cuni 4238   iotacio 5540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-sn 4021  df-pr 4023  df-uni 4239  df-iota 5542
This theorem is referenced by: (None)
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