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 ∈ 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 ∈ V
2	1	elcompl 3226	. . . . 5 ⊢ (m ∈ ∼ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ↔ ¬ m ∈ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})))
3		brres 4950	. . . . . . . . . 10 ⊢ (t(1st ↾ (◡2nd “ {0c}))m ↔ (t1st m ∧ t ∈ (◡2nd “ {0c})))
4		eliniseg 5021	. . . . . . . . . . 11 ⊢ (t ∈ (◡2nd “ {0c}) ↔ t2nd 0c)
5	4	anbi2i 675	. . . . . . . . . 10 ⊢ ((t1st m ∧ t ∈ (◡2nd “ {0c})) ↔ (t1st m ∧ t2nd 0c))
6		0cex 4393	. . . . . . . . . . 11 ⊢ 0c ∈ V
7	1, 6	op1st2nd 5791	. . . . . . . . . 10 ⊢ ((t1st m ∧ t2nd 0c) ↔ t = ⟨m, 0c⟩)
8	3, 5, 7	3bitri 262	. . . . . . . . 9 ⊢ (t(1st ↾ (◡2nd “ {0c}))m ↔ t = ⟨m, 0c⟩)
9	8	rexbii 2640	. . . . . . . 8 ⊢ (∃t ∈ (◡ FullFun ↑c “ {∅})t(1st ↾ (◡2nd “ {0c}))m ↔ ∃t ∈ (◡ FullFun ↑c “ {∅})t = ⟨m, 0c⟩)
10		elima 4755	. . . . . . . 8 ⊢ (m ∈ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ↔ ∃t ∈ (◡ FullFun ↑c “ {∅})t(1st ↾ (◡2nd “ {0c}))m)
11		risset 2662	. . . . . . . 8 ⊢ (⟨m, 0c⟩ ∈ (◡ FullFun ↑c “ {∅}) ↔ ∃t ∈ (◡ FullFun ↑c “ {∅})t = ⟨m, 0c⟩)
12	9, 10, 11	3bitr4i 268	. . . . . . 7 ⊢ (m ∈ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ↔ ⟨m, 0c⟩ ∈ (◡ FullFun ↑c “ {∅}))
13		eliniseg 5021	. . . . . . 7 ⊢ (⟨m, 0c⟩ ∈ (◡ 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 ∈ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ↔ (m ↑c 0c) = ∅)
16	15	necon3bbii 2548	. . . . 5 ⊢ (¬ m ∈ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ↔ (m ↑c 0c) ≠ ∅)
17	2, 16	bitri 240	. . . 4 ⊢ (m ∈ ∼ ((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 ∈ V
20		2ndex 5113	. . . . . . . 8 ⊢ 2nd ∈ V
21	20	cnvex 5103	. . . . . . 7 ⊢ ◡2nd ∈ V
22		snex 4112	. . . . . . 7 ⊢ {0c} ∈ V
23	21, 22	imaex 4748	. . . . . 6 ⊢ (◡2nd “ {0c}) ∈ V
24	19, 23	resex 5118	. . . . 5 ⊢ (1st ↾ (◡2nd “ {0c})) ∈ V
25		ceex 6175	. . . . . . . 8 ⊢ ↑c ∈ V
26	25	fullfunex 5861	. . . . . . 7 ⊢ FullFun ↑c ∈ V
27	26	cnvex 5103	. . . . . 6 ⊢ ◡ FullFun ↑c ∈ V
28		snex 4112	. . . . . 6 ⊢ {∅} ∈ V
29	27, 28	imaex 4748	. . . . 5 ⊢ (◡ FullFun ↑c “ {∅}) ∈ V
30	24, 29	imaex 4748	. . . 4 ⊢ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ∈ V
31	30	complex 4105	. . 3 ⊢ ∼ ((1st ↾ (◡2nd “ {0c})) “ (◡ FullFun ↑c “ {∅})) ∈ V
32	18, 31	eqeltrri 2424	. 2 ⊢ {m ∣ (m ↑c 0c) ≠ ∅} ∈ 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 +c 1c) → (m ↑c 0c) = ((n +c 1c) ↑c 0c))
38	37	neeq1d 2530	. 2 ⊢ (m = (n +c 1c) → ((m ↑c 0c) ≠ ∅ ↔ ((n +c 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 ∈ NC
42		pw10 4162	. . . 4 ⊢ ℘1∅ = ∅
43		nulel0c 4423	. . . 4 ⊢ ∅ ∈ 0c
44	42, 43	eqeltri 2423	. . 3 ⊢ ℘1∅ ∈ 0c
45		ce0nnuli 6179	. . 3 ⊢ ((0c ∈ NC ∧ ℘1∅ ∈ 0c) → (0c ↑c 0c) ≠ ∅)
46	41, 44, 45	mp2an 653	. 2 ⊢ (0c ↑c 0c) ≠ ∅
47		nnnc 6147	. . 3 ⊢ (n ∈ Nn → n ∈ NC )
48		1cnc 6140	. . . . . 6 ⊢ 1c ∈ NC
49		0ex 4111	. . . . . . . 8 ⊢ ∅ ∈ V
50	49	pw1sn 4166	. . . . . . 7 ⊢ ℘1{∅} = {{∅}}
51	28	snel1c 4141	. . . . . . 7 ⊢ {{∅}} ∈ 1c
52	50, 51	eqeltri 2423	. . . . . 6 ⊢ ℘1{∅} ∈ 1c
53		ce0nnuli 6179	. . . . . 6 ⊢ ((1c ∈ NC ∧ ℘1{∅} ∈ 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 ∈ NC ∧ 1c ∈ NC ) → (((n +c 1c) ↑c 0c) ≠ ∅ ↔ ((n ↑c 0c) ≠ ∅ ∧ (1c ↑c 0c) ≠ ∅)))
57	48, 56	mpan2 652	. . . 4 ⊢ (n ∈ NC → (((n +c 1c) ↑c 0c) ≠ ∅ ↔ ((n ↑c 0c) ≠ ∅ ∧ (1c ↑c 0c) ≠ ∅)))
58	55, 57	syl5ibr 212	. . 3 ⊢ (n ∈ NC → ((n ↑c 0c) ≠ ∅ → ((n +c 1c) ↑c 0c) ≠ ∅))
59	47, 58	syl 15	. 2 ⊢ (n ∈ Nn → ((n ↑c 0c) ≠ ∅ → ((n +c 1c) ↑c 0c) ≠ ∅))
60	32, 34, 36, 38, 40, 46, 59	finds 4412	1 ⊢ (N ∈ Nn → (N ↑c 0c) ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2517  ∃wrex 2616  Vcvv 2860   ∼ ccompl 3206  ∅c0 3551  {csn 3738  1_cc1c 4135  ℘₁cpw1 4136   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376  ⟨cop 4562   class class class wbr 4640  1^st c1st 4718   “ cima 4723  ^◡ccnv 4772   ↾ cres 4775  2^nd c2nd 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