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Theorem bj-nuliotaALT 31694
Description: Alternate proof of bj-nuliota 31693. Note that this alternate proof uses the fact that  iota x ph evaluates to  (/) when there is no  x satisfying  ph (iotanul 5568). 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 3766 . 2  |-  (/)  C_  ( iota x A. y  -.  y  e.  x )
2 iotassuni 5569 . . 3  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  U. { x  | 
A. y  -.  y  e.  x }
3 eq0 3738 . . . . . . 7  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
43bicomi 207 . . . . . 6  |-  ( A. y  -.  y  e.  x  <->  x  =  (/) )
54abbii 2587 . . . . 5  |-  { x  |  A. y  -.  y  e.  x }  =  {
x  |  x  =  (/) }
65unieqi 4199 . . . 4  |-  U. {
x  |  A. y  -.  y  e.  x }  =  U. { x  |  x  =  (/) }
7 df-sn 3960 . . . . . 6  |-  { (/) }  =  { x  |  x  =  (/) }
87eqcomi 2480 . . . . 5  |-  { x  |  x  =  (/) }  =  { (/) }
98unieqi 4199 . . . 4  |-  U. {
x  |  x  =  (/) }  =  U. { (/)
}
10 0ex 4528 . . . . 5  |-  (/)  e.  _V
1110unisn 4205 . . . 4  |-  U. { (/)
}  =  (/)
126, 9, 113eqtri 2497 . . 3  |-  U. {
x  |  A. y  -.  y  e.  x }  =  (/)
132, 12sseqtri 3450 . 2  |-  ( iota
x A. y  -.  y  e.  x ) 
C_  (/)
141, 13eqssi 3434 1  |-  (/)  =  ( iota x A. y  -.  y  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1450    = wceq 1452   {cab 2457   (/)c0 3722   {csn 3959   U.cuni 4190   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553
This theorem is referenced by: (None)
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