Theorem dfom4 9606

Description: A simplification of df-om 7851 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 9605	. 2 ⊢ (𝑥 ∈ ω ↔ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))
2	1	eqabi 2900	1 ⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}

Syntax hints:   → wi 4  ∀wal 1561   = wceq 1563  {cab 2743  Lim wlim 6351  ωcom 7850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-om 7851

This theorem is referenced by: (None)