Theorem ce0nnulb 6183

Description: Cardinal exponentiation is nonempty iff the two sets raised to zero are nonempty. Theorem XI.2.47 of [Rosser] p. 384. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
ce0nnulb	|- ((N e. NC /\ M e. NC ) -> (((N ↑c 0c) =/= (/) /\ (M ↑c 0c) =/= (/)) <-> (N ↑c M) =/= (/)))

Proof of Theorem ce0nnulb

Dummy variables a b g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ce0nnul 6178	. . . 4 |- (N e. NC -> ((N ↑c 0c) =/= (/) <-> E.a~P1 a e. N))
2		ce0nnul 6178	. . . 4 |- (M e. NC -> ((M ↑c 0c) =/= (/) <-> E.b~P1 b e. M))
3	1, 2	bi2anan9 843	. . 3 |- ((N e. NC /\ M e. NC ) -> (((N ↑c 0c) =/= (/) /\ (M ↑c 0c) =/= (/)) <-> (E.a~P1 a e. N /\ E.b~P1 b e. M)))
4		eeanv 1913	. . 3 |- (E.aE.b(~P1 a e. N /\ ~P1 b e. M) <-> (E.a~P1 a e. N /\ E.b~P1 b e. M))
5	3, 4	syl6bbr 254	. 2 |- ((N e. NC /\ M e. NC ) -> (((N ↑c 0c) =/= (/) /\ (M ↑c 0c) =/= (/)) <-> E.aE.b(~P1 a e. N /\ ~P1 b e. M)))
6		ncseqnc 6129	. . . . . 6 |- (N e. NC -> (N = Nc ~P1 a <-> ~P1 a e. N))
7		ncseqnc 6129	. . . . . 6 |- (M e. NC -> (M = Nc ~P1 b <-> ~P1 b e. M))
8	6, 7	bi2anan9 843	. . . . 5 |- ((N e. NC /\ M e. NC ) -> ((N = Nc ~P1 a /\ M = Nc ~P1 b) <-> (~P1 a e. N /\ ~P1 b e. M)))
9		vex 2863	. . . . . . . 8 |- a e. _V
10		vex 2863	. . . . . . . 8 |- b e. _V
11	9, 10	cenc 6182	. . . . . . 7 |- ( Nc ~P1 a ↑c Nc ~P1 b) = Nc (a ^m b)
12		ovex 5552	. . . . . . . . 9 |- (a ^m b) e. _V
13	12	ncid 6124	. . . . . . . 8 |- (a ^m b) e. Nc (a ^m b)
14		ne0i 3557	. . . . . . . 8 |- ((a ^m b) e. Nc (a ^m b) -> Nc (a ^m b) =/= (/))
15	13, 14	ax-mp 5	. . . . . . 7 |- Nc (a ^m b) =/= (/)
16	11, 15	eqnetri 2534	. . . . . 6 |- ( Nc ~P1 a ↑c Nc ~P1 b) =/= (/)
17		oveq12 5533	. . . . . . 7 |- ((N = Nc ~P1 a /\ M = Nc ~P1 b) -> (N ↑c M) = ( Nc ~P1 a ↑c Nc ~P1 b))
18	17	neeq1d 2530	. . . . . 6 |- ((N = Nc ~P1 a /\ M = Nc ~P1 b) -> ((N ↑c M) =/= (/) <-> ( Nc ~P1 a ↑c Nc ~P1 b) =/= (/)))
19	16, 18	mpbiri 224	. . . . 5 |- ((N = Nc ~P1 a /\ M = Nc ~P1 b) -> (N ↑c M) =/= (/))
20	8, 19	syl6bir 220	. . . 4 |- ((N e. NC /\ M e. NC ) -> ((~P1 a e. N /\ ~P1 b e. M) -> (N ↑c M) =/= (/)))
21	20	exlimdvv 1637	. . 3 |- ((N e. NC /\ M e. NC ) -> (E.aE.b(~P1 a e. N /\ ~P1 b e. M) -> (N ↑c M) =/= (/)))
22		n0 3560	. . . 4 |- ((N ↑c M) =/= (/) <-> E.g g e. (N ↑c M))
23		elce 6176	. . . . . 6 |- ((N e. NC /\ M e. NC ) -> (g e. (N ↑c M) <-> E.aE.b(~P1 a e. N /\ ~P1 b e. M /\ g ~~ (a ^m b))))
24		3simpa 952	. . . . . . 7 |- ((~P1 a e. N /\ ~P1 b e. M /\ g ~~ (a ^m b)) -> (~P1 a e. N /\ ~P1 b e. M))
25	24	2eximi 1577	. . . . . 6 |- (E.aE.b(~P1 a e. N /\ ~P1 b e. M /\ g ~~ (a ^m b)) -> E.aE.b(~P1 a e. N /\ ~P1 b e. M))
26	23, 25	syl6bi 219	. . . . 5 |- ((N e. NC /\ M e. NC ) -> (g e. (N ↑c M) -> E.aE.b(~P1 a e. N /\ ~P1 b e. M)))
27	26	exlimdv 1636	. . . 4 |- ((N e. NC /\ M e. NC ) -> (E.g g e. (N ↑c M) -> E.aE.b(~P1 a e. N /\ ~P1 b e. M)))
28	22, 27	syl5bi 208	. . 3 |- ((N e. NC /\ M e. NC ) -> ((N ↑c M) =/= (/) -> E.aE.b(~P1 a e. N /\ ~P1 b e. M)))
29	21, 28	impbid 183	. 2 |- ((N e. NC /\ M e. NC ) -> (E.aE.b(~P1 a e. N /\ ~P1 b e. M) <-> (N ↑c M) =/= (/)))
30	5, 29	bitrd 244	1 |- ((N e. NC /\ M e. NC ) -> (((N ↑c 0c) =/= (/) /\ (M ↑c 0c) =/= (/)) <-> (N ↑c M) =/= (/)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  (/)c0 3551  ~P₁ cpw1 4136  0_cc0c 4375   class class class wbr 4640  (class class class)co 5526   ^m cmap 6000   ~~ cen 6029   NC cncs 6089   Nc cnc 6092   ↑_c cce 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-ce 6107

This theorem is referenced by:  ceclb  6184