Theorem limomss 2378

Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity.

Assertion

Ref	Expression
limomss	⊢ (Lim A → ω ⊆ A)

Proof of Theorem limomss

Step	Hyp	Ref	Expression
1		limord 2283	. 2 ⊢ (Lim A → Ord A)
2		ordeleqon 2241	. . 3 ⊢ (Ord A ↔ (A ∈ On ∨ A = On))
3		limeq 2211	. . . . . . . . . . 11 ⊢ (y = A → (Lim y ↔ Lim A))
4		eleq2 1150	. . . . . . . . . . 11 ⊢ (y = A → (x ∈ y ↔ x ∈ A))
5	3, 4	imbi12d 474	. . . . . . . . . 10 ⊢ (y = A → ((Lim y → x ∈ y) ↔ (Lim A → x ∈ A)))
6	5	cla4gv 1396	. . . . . . . . 9 ⊢ (A ∈ On → (∀y(Lim y → x ∈ y) → (Lim A → x ∈ A)))
7		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
8	7	elom 2375	. . . . . . . . . 10 ⊢ (x ∈ ω ↔ (Ord x ∧ ∀y(Lim y → x ∈ y)))
9	8	pm3.27bd 263	. . . . . . . . 9 ⊢ (x ∈ ω → ∀y(Lim y → x ∈ y))
10	6, 9	syl5 22	. . . . . . . 8 ⊢ (A ∈ On → (x ∈ ω → (Lim A → x ∈ A)))
11	10	com23 32	. . . . . . 7 ⊢ (A ∈ On → (Lim A → (x ∈ ω → x ∈ A)))
12	11	imp 277	. . . . . 6 ⊢ ((A ∈ On ∧ Lim A) → (x ∈ ω → x ∈ A))
13	12	ssrdv 1509	. . . . 5 ⊢ ((A ∈ On ∧ Lim A) → ω ⊆ A)
14	13	exp 291	. . . 4 ⊢ (A ∈ On → (Lim A → ω ⊆ A))
15		omsson 2377	. . . . . 6 ⊢ ω ⊆ On
16		sseq2 1522	. . . . . 6 ⊢ (A = On → (ω ⊆ A ↔ ω ⊆ On))
17	15, 16	mpbiri 169	. . . . 5 ⊢ (A = On → ω ⊆ A)
18	17	a1d 14	. . . 4 ⊢ (A = On → (Lim A → ω ⊆ A))
19	14, 18	jaoi 275	. . 3 ⊢ ((A ∈ On ∨ A = On) → (Lim A → ω ⊆ A))
20	2, 19	sylbi 174	. 2 ⊢ (Ord A → (Lim A → ω ⊆ A))
21	1, 20	mpcom 49	1 ⊢ (Lim A → ω ⊆ A)

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Ord word 2198  Oncon0 2199  Lim wlim 2200  ωcom 2372

This theorem is referenced by:  cardlim 3657

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  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-tp 1814  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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