Theorem nsuceq0 6239

Description: No successor is empty. (Contributed by NM, 3-Apr-1995.)

Assertion

Ref	Expression
nsuceq0	⊢ suc 𝐴 ≠ ∅

Proof of Theorem nsuceq0

Step	Hyp	Ref	Expression
1		noel 4247	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
2		sucidg 6237	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		eleq2 2878	. . . . 5 ⊢ (suc 𝐴 = ∅ → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ ∅))
4	2, 3	syl5ibcom 248	. . . 4 ⊢ (𝐴 ∈ V → (suc 𝐴 = ∅ → 𝐴 ∈ ∅))
5	1, 4	mtoi 202	. . 3 ⊢ (𝐴 ∈ V → ¬ suc 𝐴 = ∅)
6		0ex 5175	. . . . . 6 ⊢ ∅ ∈ V
7		eleq1 2877	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ∈ V ↔ ∅ ∈ V))
8	6, 7	mpbiri 261	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 ∈ V)
9	8	con3i 157	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 = ∅)
10		sucprc 6234	. . . . 5 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
11	10	eqeq1d 2800	. . . 4 ⊢ (¬ 𝐴 ∈ V → (suc 𝐴 = ∅ ↔ 𝐴 = ∅))
12	9, 11	mtbird 328	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ suc 𝐴 = ∅)
13	5, 12	pm2.61i 185	. 2 ⊢ ¬ suc 𝐴 = ∅
14	13	neir 2990	1 ⊢ suc 𝐴 ≠ ∅

Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441  ∅c0 4243  suc csuc 6161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-suc 6165

This theorem is referenced by:  0elsuc  7530  peano3  7583  2on0  8096  oelim2  8204  limenpsi  8676  enp1i  8737  findcard2  8742  fseqdom  9437  dfac12lem2  9555  cfsuc  9668  cfpwsdom  9995  rankcf  10188  dfrdg2  33153  nosgnn0  33278  sltsolem1  33293  dfrdg4  33525  dfsucon  40231  ensucne0  40237  ensucne0OLD  40238