Theorem elom 2375

Description: Membership in omega. The hypothesis can be eliminated if we assume the Axiom of Infinity; see elom3 3478.

Hypothesis

Ref	Expression
elom.1	⊢ A ∈ V

Assertion

Ref	Expression
elom	⊢ (A ∈ ω ↔ (Ord A ∧ ∀x(Lim x → A ∈ x)))

Distinct variable group(s):   x,A

Proof of Theorem elom

Step	Hyp	Ref	Expression
1		elom.1	. 2 ⊢ A ∈ V
2		ordeq 2206	. . 3 ⊢ (y = A → (Ord y ↔ Ord A))
3		eleq1 1149	. . . . 5 ⊢ (y = A → (y ∈ x ↔ A ∈ x))
4	3	imbi2d 464	. . . 4 ⊢ (y = A → ((Lim x → y ∈ x) ↔ (Lim x → A ∈ x)))
5	4	bialdv 935	. . 3 ⊢ (y = A → (∀x(Lim x → y ∈ x) ↔ ∀x(Lim x → A ∈ x)))
6	2, 5	anbi12d 476	. 2 ⊢ (y = A → ((Ord y ∧ ∀x(Lim x → y ∈ x)) ↔ (Ord A ∧ ∀x(Lim x → A ∈ x))))
7		df-om 2373	. 2 ⊢ ω = {y∣(Ord y ∧ ∀x(Lim x → y ∈ x))}
8	1, 6, 7	elab2 1419	1 ⊢ (A ∈ ω ↔ (Ord A ∧ ∀x(Lim x → A ∈ x)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348  Ord word 2198  Lim wlim 2200  ωcom 2372

This theorem is referenced by:  elomg 2376  omsson 2377  limomss 2378  ordom 2382

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-16 922  ax-17 925  ax-ext 1074

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-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-om 2373

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