Theorem nlimsucg 7689

Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
nlimsucg	⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

Proof of Theorem nlimsucg

Step	Hyp	Ref	Expression
1		limord 6325	. . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 7661	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 233	. . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 6326	. . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5		ordunisuc 7679	. . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
6	5	eqeq2d 2749	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
7		ordirr 6284	. . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
8		eleq2 2827	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
9	8	notbid 318	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
10	7, 9	syl5ibrcom 246	. . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11		sucidg 6344	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
12	11	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
13	10, 12	syl6 35	. . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
14	6, 13	sylbid 239	. . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
15	3, 4, 14	sylc 65	. 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
16	15	con2i 139	1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2106  ∪ cuni 4839  Ord word 6265  Lim wlim 6267  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272

This theorem is referenced by:  tz7.44-2  8238  rankxpsuc  9640  dfrdg2  33771  scutbdaybnd2lim  34011  dfrdg4  34253  dfsucon  41130