Theorem nsuceq0 2306

Description: No successor is empty.

Assertion

Ref	Expression
nsuceq0	⊢ ¬ suc A = ∅

Proof of Theorem nsuceq0

Step	Hyp	Ref	Expression
1		noel 1711	. . 3 ⊢ ¬ A ∈ ∅
2		eleq2 1150	. . . . 5 ⊢ (suc A = ∅ → (A ∈ suc A ↔ A ∈ ∅))
3		sucidg 2305	. . . . 5 ⊢ (A ∈ V → A ∈ suc A)
4	2, 3	syl5bi 183	. . . 4 ⊢ (suc A = ∅ → (A ∈ V → A ∈ ∅))
5	4	com12 13	. . 3 ⊢ (A ∈ V → (suc A = ∅ → A ∈ ∅))
6	1, 5	mtoi 94	. 2 ⊢ (A ∈ V → ¬ suc A = ∅)
7		sucprc 2297	. . . . . 6 ⊢ (¬ A ∈ V → suc A = A)
8	7	cleq1d 1109	. . . . 5 ⊢ (¬ A ∈ V → (suc A = ∅ ↔ A = ∅))
9		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
10		eleq1 1149	. . . . . 6 ⊢ (A = ∅ → (A ∈ V ↔ ∅ ∈ V))
11	9, 10	mpbiri 169	. . . . 5 ⊢ (A = ∅ → A ∈ V)
12	8, 11	syl6bi 187	. . . 4 ⊢ (¬ A ∈ V → (suc A = ∅ → A ∈ V))
13	12	con3d 87	. . 3 ⊢ (¬ A ∈ V → (¬ A ∈ V → ¬ suc A = ∅))
14	13	pm2.43i 58	. 2 ⊢ (¬ A ∈ V → ¬ suc A = ∅)
15	6, 14	pm2.61i 110	1 ⊢ ¬ suc A = ∅

Syntax hints:  ¬ wn 1   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  suc csuc 2201

This theorem is referenced by:  0elsuc 2340  peano3 2392  tz7.44-2 2967  limenpsi 3400  cfsuc 3709  1pi 3805

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812  df-suc 2205

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