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Theorem nncex 4396

Description: The class of all finite cardinals is a set. (Contributed by SF, 14-Jan-2015.)

**Assertion**
Ref	Expression
nncex	Nn

**Proof of Theorem nncex**
Step	Hyp	Ref	Expression
1		dfnnc2 4395	. 2 Nn 0_c S_k S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c_k1_c
2		setswithex 4322	. . . 4 0_c
3		ssetkex 4294	. . . . . 6 S_k
4		addcexlem 4382	. . . . . . . . . 10 Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c
5		1cex 4142	. . . . . . . . . . . 12 1_c
6	5	pw1ex 4303	. . . . . . . . . . 11 ₁ 1_c
7	6	pw1ex 4303	. . . . . . . . . 10 ₁ ₁ 1_c
8	4, 7	imakex 4300	. . . . . . . . 9 Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c
9	8	imagekex 4312	. . . . . . . 8 Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c
10	9	sikex 4297	. . . . . . 7 SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c
11	3, 10	cokex 4310	. . . . . 6 S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c
12	3, 11	difex 4107	. . . . 5 S_k S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c
13	12, 5	imakex 4300	. . . 4 S_k S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c_k1_c
14	2, 13	difex 4107	. . 3 0_c S_k S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c_k1_c
15	14	intex 4320	. 2 0_c S_k S_k _k SI_k Image_k Ins3_k ∼ Ins3_k S_k Ins2_k S_k _k₁ ₁ 1_c Ins2_k Ins2_k S_k Ins2_k Ins3_k S_k Ins3_k SI_k SI_k S_k _k₁ ₁ ₁ ₁ 1_c_k₁ ₁ 1_c_k1_c
16	1, 15	eqeltri 2423	1 Nn

Colors of variables: wff setvar class

Syntax hints:

wcel 1710

cab 2339

cvv 2859 ∼ ccompl 3205

cdif 3206

cun 3207

cin 3208

csymdif 3209

cint 3926 1_cc1c 4134

₁ cpw1 4135 Ins2_k cins2k 4176 Ins3_k cins3k 4177

_kcimak 4179

_k ccomk 4180 SI_k csik 4181 Image_kcimagek 4182 S_k cssetk 4183 Nn cnnc 4373 0_cc0c 4374

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-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-addc 4378 df-nnc 4379

This theorem is referenced by: finex 4397 peano5 4409 nnc0suc 4412 nncaddccl 4419 nndisjeq 4429 ltfinex 4464 ncfinraiselem2 4480 ncfinlowerlem1 4482 tfinrelkex 4487 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnpweqlem1 4522 srelkex 4525 tfinnnlem1 4533 vfinspnn 4541 phiexg 4571 opexg 4587 proj1exg 4591 proj2exg 4592 phialllem1 4616 phialllem2 4617 setconslem5 4735 1stex 4739 swapex 4742 nclennlem1 6248 nmembers1lem1 6268 nncdiv3lem2 6276 nnc3n3p1 6278 nchoicelem16 6304 frecxp 6314

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