Theorem nulnnn 4557

Description: The empty class is not a natural. Theorem X.1.65 of [Rosser] p. 536. (Contributed by SF, 20-Jan-2015.)

Assertion

Ref	Expression
nulnnn	|- -. (/) e. Nn

Proof of Theorem nulnnn

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

Step	Hyp	Ref	Expression
1		complab 3525	. . . . . . 7 |- ∼ {n | n = (/)} = {n | -. n = (/)}
2		df-sn 3742	. . . . . . . 8 |- {(/)} = {n | n = (/)}
3	2	compleqi 3245	. . . . . . 7 |- ∼ {(/)} = ∼ {n | n = (/)}
4		df-ne 2519	. . . . . . . 8 |- (n =/= (/) <-> -. n = (/))
5	4	abbii 2466	. . . . . . 7 |- {n | n =/= (/)} = {n | -. n = (/)}
6	1, 3, 5	3eqtr4ri 2384	. . . . . 6 |- {n | n =/= (/)} = ∼ {(/)}
7		snex 4112	. . . . . . 7 |- {(/)} e. _V
8	7	complex 4105	. . . . . 6 |- ∼ {(/)} e. _V
9	6, 8	eqeltri 2423	. . . . 5 |- {n | n =/= (/)} e. _V
10		neeq1 2525	. . . . 5 |- (n = 0c -> (n =/= (/) <-> 0c =/= (/)))
11		neeq1 2525	. . . . 5 |- (n = m -> (n =/= (/) <-> m =/= (/)))
12		neeq1 2525	. . . . 5 |- (n = (m +o 1c) -> (n =/= (/) <-> (m +o 1c) =/= (/)))
13		neeq1 2525	. . . . 5 |- (n = x -> (n =/= (/) <-> x =/= (/)))
14		nulel0c 4423	. . . . . 6 |- (/) e. 0c
15		ne0i 3557	. . . . . 6 |- ((/) e. 0c -> 0c =/= (/))
16	14, 15	ax-mp 5	. . . . 5 |- 0c =/= (/)
17		n0 3560	. . . . . 6 |- (m =/= (/) <-> E.a a e. m)
18		vinf 4556	. . . . . . . . . . . . . . 15 |- -. _V e. Fin
19		elunii 3897	. . . . . . . . . . . . . . . . . 18 |- ((_V e. m /\ m e. Nn ) -> _V e. U. Nn )
20	19	ancoms 439	. . . . . . . . . . . . . . . . 17 |- ((m e. Nn /\ _V e. m) -> _V e. U. Nn )
21		df-fin 4381	. . . . . . . . . . . . . . . . 17 |- Fin = U. Nn
22	20, 21	syl6eleqr 2444	. . . . . . . . . . . . . . . 16 |- ((m e. Nn /\ _V e. m) -> _V e. Fin )
23	22	ex 423	. . . . . . . . . . . . . . 15 |- (m e. Nn -> (_V e. m -> _V e. Fin ))
24	18, 23	mtoi 169	. . . . . . . . . . . . . 14 |- (m e. Nn -> -. _V e. m)
25		eleq1 2413	. . . . . . . . . . . . . . 15 |- (a = _V -> (a e. m <-> _V e. m))
26	25	notbid 285	. . . . . . . . . . . . . 14 |- (a = _V -> (-. a e. m <-> -. _V e. m))
27	24, 26	syl5ibrcom 213	. . . . . . . . . . . . 13 |- (m e. Nn -> (a = _V -> -. a e. m))
28	27	necon2ad 2565	. . . . . . . . . . . 12 |- (m e. Nn -> (a e. m -> a =/= _V))
29	28	imp 418	. . . . . . . . . . 11 |- ((m e. Nn /\ a e. m) -> a =/= _V)
30		compleqb 3544	. . . . . . . . . . . 12 |- (a = _V <-> ∼ a = ∼ _V)
31	30	necon3bii 2549	. . . . . . . . . . 11 |- (a =/= _V <-> ∼ a =/= ∼ _V)
32	29, 31	sylib 188	. . . . . . . . . 10 |- ((m e. Nn /\ a e. m) -> ∼ a =/= ∼ _V)
33		complV 4071	. . . . . . . . . . 11 |- ∼ _V = (/)
34	33	neeq2i 2528	. . . . . . . . . 10 |- ( ∼ a =/= ∼ _V <-> ∼ a =/= (/))
35	32, 34	sylib 188	. . . . . . . . 9 |- ((m e. Nn /\ a e. m) -> ∼ a =/= (/))
36		n0 3560	. . . . . . . . . 10 |- ( ∼ a =/= (/) <-> E.x x e. ∼ a)
37		vex 2863	. . . . . . . . . . . . . . 15 |- x e. _V
38	37	elcompl 3226	. . . . . . . . . . . . . 14 |- (x e. ∼ a <-> -. x e. a)
39	37	elsuci 4415	. . . . . . . . . . . . . . 15 |- ((a e. m /\ -. x e. a) -> (a u. {x}) e. (m +o 1c))
40		ne0i 3557	. . . . . . . . . . . . . . 15 |- ((a u. {x}) e. (m +o 1c) -> (m +o 1c) =/= (/))
41	39, 40	syl 15	. . . . . . . . . . . . . 14 |- ((a e. m /\ -. x e. a) -> (m +o 1c) =/= (/))
42	38, 41	sylan2b 461	. . . . . . . . . . . . 13 |- ((a e. m /\ x e. ∼ a) -> (m +o 1c) =/= (/))
43	42	ex 423	. . . . . . . . . . . 12 |- (a e. m -> (x e. ∼ a -> (m +o 1c) =/= (/)))
44	43	adantl 452	. . . . . . . . . . 11 |- ((m e. Nn /\ a e. m) -> (x e. ∼ a -> (m +o 1c) =/= (/)))
45	44	exlimdv 1636	. . . . . . . . . 10 |- ((m e. Nn /\ a e. m) -> (E.x x e. ∼ a -> (m +o 1c) =/= (/)))
46	36, 45	syl5bi 208	. . . . . . . . 9 |- ((m e. Nn /\ a e. m) -> ( ∼ a =/= (/) -> (m +o 1c) =/= (/)))
47	35, 46	mpd 14	. . . . . . . 8 |- ((m e. Nn /\ a e. m) -> (m +o 1c) =/= (/))
48	47	ex 423	. . . . . . 7 |- (m e. Nn -> (a e. m -> (m +o 1c) =/= (/)))
49	48	exlimdv 1636	. . . . . 6 |- (m e. Nn -> (E.a a e. m -> (m +o 1c) =/= (/)))
50	17, 49	syl5bi 208	. . . . 5 |- (m e. Nn -> (m =/= (/) -> (m +o 1c) =/= (/)))
51	9, 10, 11, 12, 13, 16, 50	finds 4412	. . . 4 |- (x e. Nn -> x =/= (/))
52	51	neneqd 2533	. . 3 |- (x e. Nn -> -. x = (/))
53	52	nrex 2717	. 2 |- -. E.x e. Nn x = (/)
54		risset 2662	. 2 |- ((/) e. Nn <-> E.x e. Nn x = (/))
55	53, 54	mtbir 290	1 |- -. (/) e. Nn

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  E.wrex 2616  _Vcvv 2860   ∼ ccompl 3206   u. cun 3208  (/)c0 3551  {csn 3738  U.cuni 3892  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +o cplc 4376   Fin cfin 4377

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

This theorem is referenced by:  peano4  4558  addccan2  4560  nntccl  6171