Theorem ce0nn 6181

Description: A natural raised to cardinal zero is nonempty. Theorem XI.2.44 of [Rosser] p. 383. (Contributed by SF, 9-Mar-2015.)

Assertion

Ref	Expression
ce0nn	|- (N e. Nn -> (N ↑c 0c) =/= (/))

Proof of Theorem ce0nn

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . 6 |- m e. _V
2	1	elcompl 3226	. . . . 5 |- (m e. ∼ ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> -. m e. ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})))
3		brres 4950	. . . . . . . . . 10 |- (t(1st |` (`'2nd"{0c}))m <-> (t1stm /\ t e. (`'2nd"{0c})))
4		eliniseg 5021	. . . . . . . . . . 11 |- (t e. (`'2nd"{0c}) <-> t2nd0c)
5	4	anbi2i 675	. . . . . . . . . 10 |- ((t1stm /\ t e. (`'2nd"{0c})) <-> (t1stm /\ t2nd0c))
6		0cex 4393	. . . . . . . . . . 11 |- 0c e. _V
7	1, 6	op1st2nd 5791	. . . . . . . . . 10 |- ((t1stm /\ t2nd0c) <-> t = <.m, 0c>.)
8	3, 5, 7	3bitri 262	. . . . . . . . 9 |- (t(1st |` (`'2nd"{0c}))m <-> t = <.m, 0c>.)
9	8	rexbii 2640	. . . . . . . 8 |- (E.t e. (`' FullFun ↑c "{(/)})t(1st |` (`'2nd"{0c}))m <-> E.t e. (`' FullFun ↑c "{(/)})t = <.m, 0c>.)
10		elima 4755	. . . . . . . 8 |- (m e. ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> E.t e. (`' FullFun ↑c "{(/)})t(1st |` (`'2nd"{0c}))m)
11		risset 2662	. . . . . . . 8 |- (<.m, 0c>. e. (`' FullFun ↑c "{(/)}) <-> E.t e. (`' FullFun ↑c "{(/)})t = <.m, 0c>.)
12	9, 10, 11	3bitr4i 268	. . . . . . 7 |- (m e. ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> <.m, 0c>. e. (`' FullFun ↑c "{(/)}))
13		eliniseg 5021	. . . . . . 7 |- (<.m, 0c>. e. (`' FullFun ↑c "{(/)}) <-> <.m, 0c>. FullFun ↑c (/))
14	1, 6	brfullfunop 5868	. . . . . . 7 |- (<.m, 0c>. FullFun ↑c (/) <-> (m ↑c 0c) = (/))
15	12, 13, 14	3bitri 262	. . . . . 6 |- (m e. ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> (m ↑c 0c) = (/))
16	15	necon3bbii 2548	. . . . 5 |- (-. m e. ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> (m ↑c 0c) =/= (/))
17	2, 16	bitri 240	. . . 4 |- (m e. ∼ ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) <-> (m ↑c 0c) =/= (/))
18	17	eqabi 2465	. . 3 |- ∼ ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) = {m | (m ↑c 0c) =/= (/)}
19		1stex 4740	. . . . . 6 |- 1st e. _V
20		2ndex 5113	. . . . . . . 8 |- 2nd e. _V
21	20	cnvex 5103	. . . . . . 7 |- `'2nd e. _V
22		snex 4112	. . . . . . 7 |- {0c} e. _V
23	21, 22	imaex 4748	. . . . . 6 |- (`'2nd"{0c}) e. _V
24	19, 23	resex 5118	. . . . 5 |- (1st |` (`'2nd"{0c})) e. _V
25		ceex 6175	. . . . . . . 8 |- ↑c e. _V
26	25	fullfunex 5861	. . . . . . 7 |- FullFun ↑c e. _V
27	26	cnvex 5103	. . . . . 6 |- `' FullFun ↑c e. _V
28		snex 4112	. . . . . 6 |- {(/)} e. _V
29	27, 28	imaex 4748	. . . . 5 |- (`' FullFun ↑c "{(/)}) e. _V
30	24, 29	imaex 4748	. . . 4 |- ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) e. _V
31	30	complex 4105	. . 3 |- ∼ ((1st |` (`'2nd"{0c}))"(`' FullFun ↑c "{(/)})) e. _V
32	18, 31	eqeltrri 2424	. 2 |- {m | (m ↑c 0c) =/= (/)} e. _V
33		oveq1 5531	. . 3 |- (m = 0c -> (m ↑c 0c) = (0c ↑c 0c))
34	33	neeq1d 2530	. 2 |- (m = 0c -> ((m ↑c 0c) =/= (/) <-> (0c ↑c 0c) =/= (/)))
35		oveq1 5531	. . 3 |- (m = n -> (m ↑c 0c) = (n ↑c 0c))
36	35	neeq1d 2530	. 2 |- (m = n -> ((m ↑c 0c) =/= (/) <-> (n ↑c 0c) =/= (/)))
37		oveq1 5531	. . 3 |- (m = (n +o 1c) -> (m ↑c 0c) = ((n +o 1c) ↑c 0c))
38	37	neeq1d 2530	. 2 |- (m = (n +o 1c) -> ((m ↑c 0c) =/= (/) <-> ((n +o 1c) ↑c 0c) =/= (/)))
39		oveq1 5531	. . 3 |- (m = N -> (m ↑c 0c) = (N ↑c 0c))
40	39	neeq1d 2530	. 2 |- (m = N -> ((m ↑c 0c) =/= (/) <-> (N ↑c 0c) =/= (/)))
41		0cnc 6139	. . 3 |- 0c e. NC
42		pw10 4162	. . . 4 |- ~P1 (/) = (/)
43		nulel0c 4423	. . . 4 |- (/) e. 0c
44	42, 43	eqeltri 2423	. . 3 |- ~P1 (/) e. 0c
45		ce0nnuli 6179	. . 3 |- ((0c e. NC /\ ~P1 (/) e. 0c) -> (0c ↑c 0c) =/= (/))
46	41, 44, 45	mp2an 653	. 2 |- (0c ↑c 0c) =/= (/)
47		nnnc 6147	. . 3 |- (n e. Nn -> n e. NC )
48		1cnc 6140	. . . . . 6 |- 1c e. NC
49		0ex 4111	. . . . . . . 8 |- (/) e. _V
50	49	pw1sn 4166	. . . . . . 7 |- ~P1 {(/)} = {{(/)}}
51	28	snel1c 4141	. . . . . . 7 |- {{(/)}} e. 1c
52	50, 51	eqeltri 2423	. . . . . 6 |- ~P1 {(/)} e. 1c
53		ce0nnuli 6179	. . . . . 6 |- ((1c e. NC /\ ~P1 {(/)} e. 1c) -> (1c ↑c 0c) =/= (/))
54	48, 52, 53	mp2an 653	. . . . 5 |- (1c ↑c 0c) =/= (/)
55	54	jctr 526	. . . 4 |- ((n ↑c 0c) =/= (/) -> ((n ↑c 0c) =/= (/) /\ (1c ↑c 0c) =/= (/)))
56		ce0addcnnul 6180	. . . . 5 |- ((n e. NC /\ 1c e. NC ) -> (((n +o 1c) ↑c 0c) =/= (/) <-> ((n ↑c 0c) =/= (/) /\ (1c ↑c 0c) =/= (/))))
57	48, 56	mpan2 652	. . . 4 |- (n e. NC -> (((n +o 1c) ↑c 0c) =/= (/) <-> ((n ↑c 0c) =/= (/) /\ (1c ↑c 0c) =/= (/))))
58	55, 57	syl5ibr 212	. . 3 |- (n e. NC -> ((n ↑c 0c) =/= (/) -> ((n +o 1c) ↑c 0c) =/= (/)))
59	47, 58	syl 15	. 2 |- (n e. Nn -> ((n ↑c 0c) =/= (/) -> ((n +o 1c) ↑c 0c) =/= (/)))
60	32, 34, 36, 38, 40, 46, 59	finds 4412	1 |- (N e. Nn -> (N ↑c 0c) =/= (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206  (/)c0 3551  {csn 3738  1_cc1c 4135  ~P₁ cpw1 4136   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376  <.cop 4562   class class class wbr 4640  1stc1st 4718  "cima 4723  `'ccnv 4772   |` cres 4775  2ndc2nd 4784  (class class class)co 5526   FullFun cfullfun 5768   NC cncs 6089   ↑_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-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  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:  ceclnn1  6190  nchoicelem5  6294