Theorem finxp00 33239

Description: Cartesian exponentiation of the empty set to any power is the empty set. (Contributed by ML, 24-Oct-2020.)

Assertion

Ref	Expression
finxp00	|- ( (/) ^^ ^^ N ) = (/)

Proof of Theorem finxp00

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		finxpeq2 33224	. . . 4 |- ( n = (/) -> ( (/) ^^ ^^ n ) = ( (/) ^^ ^^ (/) ) )
2	1	eqeq1d 2624	. . 3 |- ( n = (/) -> ( ( (/) ^^ ^^ n ) = (/) <-> ( (/) ^^ ^^ (/) ) = (/) ) )
3		finxpeq2 33224	. . . 4 |- ( n = m -> ( (/) ^^ ^^ n ) = ( (/) ^^ ^^ m ) )
4	3	eqeq1d 2624	. . 3 |- ( n = m -> ( ( (/) ^^ ^^ n ) = (/) <-> ( (/) ^^ ^^ m ) = (/) ) )
5		finxpeq2 33224	. . . 4 |- ( n = suc m -> ( (/) ^^ ^^ n ) = ( (/) ^^ ^^ suc m ) )
6	5	eqeq1d 2624	. . 3 |- ( n = suc m -> ( ( (/) ^^ ^^ n ) = (/) <-> ( (/) ^^ ^^ suc m ) = (/) ) )
7		finxpeq2 33224	. . . 4 |- ( n = N -> ( (/) ^^ ^^ n ) = ( (/) ^^ ^^ N ) )
8	7	eqeq1d 2624	. . 3 |- ( n = N -> ( ( (/) ^^ ^^ n ) = (/) <-> ( (/) ^^ ^^ N ) = (/) ) )
9		finxp0 33228	. . 3 |- ( (/) ^^ ^^ (/) ) = (/)
10		suceq 5790	. . . . . . . . 9 |- ( m = (/) -> suc m = suc (/) )
11		df-1o 7560	. . . . . . . . 9 |- 1o = suc (/)
12	10, 11	syl6eqr 2674	. . . . . . . 8 |- ( m = (/) -> suc m = 1o )
13		finxpeq2 33224	. . . . . . . 8 |- ( suc m = 1o -> ( (/) ^^ ^^ suc m ) = ( (/) ^^ ^^ 1o ) )
14	12, 13	syl 17	. . . . . . 7 |- ( m = (/) -> ( (/) ^^ ^^ suc m ) = ( (/) ^^ ^^ 1o ) )
15		finxp1o 33229	. . . . . . 7 |- ( (/) ^^ ^^ 1o ) = (/)
16	14, 15	syl6eq 2672	. . . . . 6 |- ( m = (/) -> ( (/) ^^ ^^ suc m ) = (/) )
17	16	adantl 482	. . . . 5 |- ( ( m e. om /\ m = (/) ) -> ( (/) ^^ ^^ suc m ) = (/) )
18		finxpsuc 33235	. . . . . 6 |- ( ( m e. om /\ m =/= (/) ) -> ( (/) ^^ ^^ suc m ) = ( ( (/) ^^ ^^ m ) X. (/) ) )
19		xp0 5552	. . . . . 6 |- ( ( (/) ^^ ^^ m ) X. (/) ) = (/)
20	18, 19	syl6eq 2672	. . . . 5 |- ( ( m e. om /\ m =/= (/) ) -> ( (/) ^^ ^^ suc m ) = (/) )
21	17, 20	pm2.61dane 2881	. . . 4 |- ( m e. om -> ( (/) ^^ ^^ suc m ) = (/) )
22	21	a1d 25	. . 3 |- ( m e. om -> ( ( (/) ^^ ^^ m ) = (/) -> ( (/) ^^ ^^ suc m ) = (/) ) )
23	2, 4, 6, 8, 9, 22	finds 7092	. 2 |- ( N e. om -> ( (/) ^^ ^^ N ) = (/) )
24		finxpnom 33238	. 2 |- ( -. N e. om -> ( (/) ^^ ^^ N ) = (/) )
25	23, 24	pm2.61i 176	1 |- ( (/) ^^ ^^ N ) = (/)

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   X. cxp 5112   suc csuc 5725   omcom 7065   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-finxp 33221

This theorem is referenced by: (None)