Theorem nlimsucg 4543

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 4373	. . . . . 6 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 4540	. . . . . 6 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 133	. . . . 5 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 4374	. . . . 5 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5	3, 4	jca 304	. . . 4 ⊢ (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
6		ordtr 4356	. . . . . . . 8 ⊢ (Ord 𝐴 → Tr 𝐴)
7		unisucg 4392	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
8	7	biimpa 294	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ Tr 𝐴) → ∪ suc 𝐴 = 𝐴)
9	6, 8	sylan2 284	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → ∪ suc 𝐴 = 𝐴)
10	9	eqeq2d 2177	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
11		ordirr 4519	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
12		eleq2 2230	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
13	12	notbid 657	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
14	11, 13	syl5ibrcom 156	. . . . . . . 8 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15		sucidg 4394	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
16	15	con3i 622	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
17	14, 16	syl6 33	. . . . . . 7 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
18	17	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
19	10, 18	sylbid 149	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
20	19	expimpd 361	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((Ord 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴) → ¬ 𝐴 ∈ 𝑉))
21	5, 20	syl5 32	. . 3 ⊢ (𝐴 ∈ 𝑉 → (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
22	21	con2d 614	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴))
23	22	pm2.43i 49	1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∪ cuni 3789  Tr wtr 4080  Ord word 4340  Lim wlim 4342  suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-ilim 4347  df-suc 4349

This theorem is referenced by: (None)