Theorem nlimsucg 4525

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 4355	. . . . . 6 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 4522	. . . . . 6 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 133	. . . . 5 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 4356	. . . . 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 4338	. . . . . . . 8 ⊢ (Ord 𝐴 → Tr 𝐴)
7		unisucg 4374	. . . . . . . . 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 2169	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
11		ordirr 4501	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
12		eleq2 2221	. . . . . . . . . 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 4376	. . . . . . . . 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 1335   ∈ wcel 2128  ∪ cuni 3772  Tr wtr 4062  Ord word 4322  Lim wlim 4324  suc csuc 4325

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4496

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-ilim 4329  df-suc 4331

This theorem is referenced by: (None)