Theorem elom 7809

Description: Membership in omega. The left conjunct can be eliminated if we assume the Axiom of Infinity; see elom3 9560. (Contributed by NM, 15-May-1994.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem elom

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2827	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2	1	imbi2d 341	. . 3 ⊢ (𝑦 = 𝐴 → ((Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ (Lim 𝑥 → 𝐴 ∈ 𝑥)))
3	2	albidv 1927	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥) ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))
4		df-om 7807	. 2 ⊢ ω = {𝑦 ∈ On ∣ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥)}
5	3, 4	elrab2 3632	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 ∈ On ∧ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  Oncon0 6310  Lim wlim 6311  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-om 7807

This theorem is referenced by:  limomss  7811  trom  7815  nnlim  7820  limom  7822  peano1  7829  1onn  8566  2onn  8568  elom3  9560  dfom5b  36138