Theorem dfom4 9696

Description: A simplification of df-om 7895 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)

Assertion

Ref	Expression
dfom4	⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfom4

Step	Hyp	Ref	Expression
1		elom3 9695	. 2 ⊢ (𝑥 ∈ ω ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
2	1	eqabi 2877	1 ⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539  {cab 2714  Lim wlim 6393  ωcom 7894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441  ax-un 7761  ax-inf2 9688

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-tr 5269  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-we 5647  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-om 7895

This theorem is referenced by: (None)