Theorem 0nelsuc 4401

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

Assertion

Ref	Expression
0nelsuc	⊢ ¬ ∅ ∈ (A +c 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 ∈ 1c ↔ ∃n m = {n})
2		vex 2863	. . . . . . . . . . . . 13 ⊢ n ∈ V
3	2	snid 3761	. . . . . . . . . . . 12 ⊢ n ∈ {n}
4		n0i 3556	. . . . . . . . . . . 12 ⊢ (n ∈ {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 ⊢ (∃n m = {n} → ¬ m = ∅)
9	1, 8	sylbi 187	. . . . . . . 8 ⊢ (m ∈ 1c → ¬ m = ∅)
10		simpr 447	. . . . . . . 8 ⊢ ((n = ∅ ∧ m = ∅) → m = ∅)
11	9, 10	nsyl 113	. . . . . . 7 ⊢ (m ∈ 1c → ¬ (n = ∅ ∧ m = ∅))
12		un00 3587	. . . . . . . . 9 ⊢ ((n = ∅ ∧ m = ∅) ↔ (n ∪ m) = ∅)
13		eqcom 2355	. . . . . . . . 9 ⊢ ((n ∪ m) = ∅ ↔ ∅ = (n ∪ m))
14	12, 13	bitri 240	. . . . . . . 8 ⊢ ((n = ∅ ∧ m = ∅) ↔ ∅ = (n ∪ m))
15	14	notbii 287	. . . . . . 7 ⊢ (¬ (n = ∅ ∧ m = ∅) ↔ ¬ ∅ = (n ∪ m))
16	11, 15	sylib 188	. . . . . 6 ⊢ (m ∈ 1c → ¬ ∅ = (n ∪ m))
17		simpr 447	. . . . . 6 ⊢ (((n ∩ m) = ∅ ∧ ∅ = (n ∪ m)) → ∅ = (n ∪ m))
18	16, 17	nsyl 113	. . . . 5 ⊢ (m ∈ 1c → ¬ ((n ∩ m) = ∅ ∧ ∅ = (n ∪ m)))
19	18	nrex 2717	. . . 4 ⊢ ¬ ∃m ∈ 1c ((n ∩ m) = ∅ ∧ ∅ = (n ∪ m))
20	19	a1i 10	. . 3 ⊢ (n ∈ A → ¬ ∃m ∈ 1c ((n ∩ m) = ∅ ∧ ∅ = (n ∪ m)))
21	20	nrex 2717	. 2 ⊢ ¬ ∃n ∈ A ∃m ∈ 1c ((n ∩ m) = ∅ ∧ ∅ = (n ∪ m))
22		eladdc 4399	. 2 ⊢ (∅ ∈ (A +c 1c) ↔ ∃n ∈ A ∃m ∈ 1c ((n ∩ m) = ∅ ∧ ∅ = (n ∪ m)))
23	21, 22	mtbir 290	1 ⊢ ¬ ∅ ∈ (A +c 1c)

Syntax hints:  ¬ wn 3   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  1_cc1c 4135   +_c 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