Theorem 0nelsuc 4401

Description: The empty class is not a member of a successor. (Contributed by SF, 14-Jan-2015.)

Assertion

Ref	Expression
0nelsuc	|- -. (/) e. (A +o 1c)

Proof of Theorem 0nelsuc

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

Step	Hyp	Ref	Expression
1		el1c 4140	. . . . . . . . 9 |- (m e. 1c <-> E.n m = {n})
2		vex 2863	. . . . . . . . . . . . 13 |- n e. _V
3	2	snid 3761	. . . . . . . . . . . 12 |- n e. {n}
4		n0i 3556	. . . . . . . . . . . 12 |- (n e. {n} -> -. {n} = (/))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 |- -. {n} = (/)
6		eqeq1 2359	. . . . . . . . . . 11 |- (m = {n} -> (m = (/) <-> {n} = (/)))
7	5, 6	mtbiri 294	. . . . . . . . . 10 |- (m = {n} -> -. m = (/))
8	7	exlimiv 1634	. . . . . . . . 9 |- (E.n m = {n} -> -. m = (/))
9	1, 8	sylbi 187	. . . . . . . 8 |- (m e. 1c -> -. m = (/))
10		simpr 447	. . . . . . . 8 |- ((n = (/) /\ m = (/)) -> m = (/))
11	9, 10	nsyl 113	. . . . . . 7 |- (m e. 1c -> -. (n = (/) /\ m = (/)))
12		un00 3587	. . . . . . . . 9 |- ((n = (/) /\ m = (/)) <-> (n u. m) = (/))
13		eqcom 2355	. . . . . . . . 9 |- ((n u. m) = (/) <-> (/) = (n u. m))
14	12, 13	bitri 240	. . . . . . . 8 |- ((n = (/) /\ m = (/)) <-> (/) = (n u. m))
15	14	notbii 287	. . . . . . 7 |- (-. (n = (/) /\ m = (/)) <-> -. (/) = (n u. m))
16	11, 15	sylib 188	. . . . . 6 |- (m e. 1c -> -. (/) = (n u. m))
17		simpr 447	. . . . . 6 |- (((n i^i m) = (/) /\ (/) = (n u. m)) -> (/) = (n u. m))
18	16, 17	nsyl 113	. . . . 5 |- (m e. 1c -> -. ((n i^i m) = (/) /\ (/) = (n u. m)))
19	18	nrex 2717	. . . 4 |- -. E.m e. 1c ((n i^i m) = (/) /\ (/) = (n u. m))
20	19	a1i 10	. . 3 |- (n e. A -> -. E.m e. 1c ((n i^i m) = (/) /\ (/) = (n u. m)))
21	20	nrex 2717	. 2 |- -. E.n e. A E.m e. 1c ((n i^i m) = (/) /\ (/) = (n u. m))
22		eladdc 4399	. 2 |- ((/) e. (A +o 1c) <-> E.n e. A E.m e. 1c ((n i^i m) = (/) /\ (/) = (n u. m)))
23	21, 22	mtbir 290	1 |- -. (/) e. (A +o 1c)

Syntax hints:  -. wn 3   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616   u. cun 3208   i^i cin 3209  (/)c0 3551  {csn 3738  1_cc1c 4135   +o cplc 4376

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 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137  df-addc 4379

This theorem is referenced by:  0cnsuc  4402  nndisjeq  4430  sfinltfin  4536