Theorem nsuceq0 3048

Description: No successor is empty.

Assertion

Ref	Expression
nsuceq0	⊢ suc A ≠ ∅

Proof of Theorem nsuceq0

Step	Hyp	Ref	Expression
1		noel 2280	. . . 4 ⊢ ¬ A ∈ ∅
2		eleq2 1532	. . . . 5 ⊢ (suc A = ∅ → (A ∈ suc A ↔ A ∈ ∅))
3		sucidg 3047	. . . . 5 ⊢ (A ∈ V → A ∈ suc A)
4	2, 3	syl5cbi 209	. . . 4 ⊢ (A ∈ V → (suc A = ∅ → A ∈ ∅))
5	1, 4	mtoi 107	. . 3 ⊢ (A ∈ V → ¬ suc A = ∅)
6		sucprc 3039	. . . . . . 7 ⊢ (¬ A ∈ V → suc A = A)
7	6	eqeq1d 1480	. . . . . 6 ⊢ (¬ A ∈ V → (suc A = ∅ ↔ A = ∅))
8		0ex 2706	. . . . . . 7 ⊢ ∅ ∈ V
9		eleq1 1531	. . . . . . 7 ⊢ (A = ∅ → (A ∈ V ↔ ∅ ∈ V))
10	8, 9	mpbiri 194	. . . . . 6 ⊢ (A = ∅ → A ∈ V)
11	7, 10	syl6bi 214	. . . . 5 ⊢ (¬ A ∈ V → (suc A = ∅ → A ∈ V))
12	11	con3d 95	. . . 4 ⊢ (¬ A ∈ V → (¬ A ∈ V → ¬ suc A = ∅))
13	12	pm2.43i 64	. . 3 ⊢ (¬ A ∈ V → ¬ suc A = ∅)
14	5, 13	pm2.61i 126	. 2 ⊢ ¬ suc A = ∅
15		df-ne 1584	. 2 ⊢ (suc A ≠ ∅ ↔ ¬ suc A = ∅)
16	14, 15	mpbir 190	1 ⊢ suc A ≠ ∅

Syntax hints:  ¬ wn 2   = wceq 954   ∈ wcel 956   ≠ wne 1582  Vcvv 1807  ∅c0 2276  suc csuc 2945

This theorem is referenced by:  0elsuc 3087  peano3 3146  tz7.44-2 3920  oelim2 4212  limenpsi 4491  cfsuc 4895

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-nul 2705

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-nul 2277  df-sn 2408  df-pr 2409  df-suc 2949

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