Theorem finxpnom 33238

Description: Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)

Assertion

Ref	Expression
finxpnom	|- ( -. N e. om -> ( U ^^ ^^ N ) = (/) )

Proof of Theorem finxpnom

Dummy variables n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) -> N e. om )
2	1	con3i 150	. . . 4 |- ( -. N e. om -> -. ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) )
3		abid 2610	. . . 4 |- ( y e. { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) } <-> ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) )
4	2, 3	sylnibr 319	. . 3 |- ( -. N e. om -> -. y e. { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) } )
5		df-finxp 33221	. . . 4 |- ( U ^^ ^^ N ) = { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) }
6	5	eleq2i 2693	. . 3 |- ( y e. ( U ^^ ^^ N ) <-> y e. { y | ( N e. om /\ (/) = ( rec ( ( n e. om , x e. _V |-> if ( ( n = 1o /\ x e. U ) , (/) , if ( x e. ( _V X. U ) , <. U. n , ( 1st ` x ) >. , <. n , x >. ) ) ) , <. N , y >. ) ` N ) ) } )
7	4, 6	sylnibr 319	. 2 |- ( -. N e. om -> -. y e. ( U ^^ ^^ N ) )
8	7	eq0rdv 3979	1 |- ( -. N e. om -> ( U ^^ ^^ N ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   X. cxp 5112   ` cfv 5888   |-> cmpt2 6652   omcom 7065   1stc1st 7166   reccrdg 7505   1oc1o 7553   ^^ ^^cfinxp 33220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-finxp 33221

This theorem is referenced by:  finxp00  33239