Theorem nnlim 2385

Description: A natural number is not a limit ordinal.

Assertion

Ref	Expression
nnlim	⊢ (A ∈ ω → ¬ Lim A)

Proof of Theorem nnlim

Step	Hyp	Ref	Expression
1		nnord 2381	. . 3 ⊢ (A ∈ ω → Ord A)
2		ordeirr 2217	. . 3 ⊢ (Ord A → ¬ A ∈ A)
3	1, 2	syl 12	. 2 ⊢ (A ∈ ω → ¬ A ∈ A)
4		elomg 2376	. . . . 5 ⊢ (A ∈ ω → (A ∈ ω ↔ (Ord A ∧ ∀x(Lim x → A ∈ x))))
5	4	ibi 449	. . . 4 ⊢ (A ∈ ω → (Ord A ∧ ∀x(Lim x → A ∈ x)))
6	5	pm3.27d 262	. . 3 ⊢ (A ∈ ω → ∀x(Lim x → A ∈ x))
7		limeq 2211	. . . . 5 ⊢ (x = A → (Lim x ↔ Lim A))
8		eleq2 1150	. . . . 5 ⊢ (x = A → (A ∈ x ↔ A ∈ A))
9	7, 8	imbi12d 474	. . . 4 ⊢ (x = A → ((Lim x → A ∈ x) ↔ (Lim A → A ∈ A)))
10	9	cla4gv 1396	. . 3 ⊢ (A ∈ ω → (∀x(Lim x → A ∈ x) → (Lim A → A ∈ A)))
11	6, 10	mpd 46	. 2 ⊢ (A ∈ ω → (Lim A → A ∈ A))
12	3, 11	mtod 95	1 ⊢ (A ∈ ω → ¬ Lim A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Ord word 2198  Lim wlim 2200  ωcom 2372

This theorem is referenced by:  omssnlim 2386  limom 2387  nnsuc 2389

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-om 2373

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