Theorem nlimsucg 4602

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 4430	. . . . . 6 ⊢ (Lim suc 𝐴 → Ord suc 𝐴)
2		ordsuc 4599	. . . . . 6 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
3	1, 2	sylibr 134	. . . . 5 ⊢ (Lim suc 𝐴 → Ord 𝐴)
4		limuni 4431	. . . . 5 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴)
5	3, 4	jca 306	. . . 4 ⊢ (Lim suc 𝐴 → (Ord 𝐴 ∧ suc 𝐴 = ∪ suc 𝐴))
6		ordtr 4413	. . . . . . . 8 ⊢ (Ord 𝐴 → Tr 𝐴)
7		unisucg 4449	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
8	7	biimpa 296	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ Tr 𝐴) → ∪ suc 𝐴 = 𝐴)
9	6, 8	sylan2 286	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → ∪ suc 𝐴 = 𝐴)
10	9	eqeq2d 2208	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴))
11		ordirr 4578	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
12		eleq2 2260	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴))
13	12	notbid 668	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴))
14	11, 13	syl5ibrcom 157	. . . . . . . 8 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
15		sucidg 4451	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴)
16	15	con3i 633	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)
17	14, 16	syl6 33	. . . . . . 7 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
18	17	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉))
19	10, 18	sylbid 150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ Ord 𝐴) → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉))
20	19	expimpd 363	. . . 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 625	. 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 104   = wceq 1364   ∈ wcel 2167  ∪ cuni 3839  Tr wtr 4131  Ord word 4397  Lim wlim 4399  suc csuc 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-ilim 4404  df-suc 4406

This theorem is referenced by: (None)