Theorem dfom5 9408

Description: ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
dfom5	⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Proof of Theorem dfom5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elom3 9406	. . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
2		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
3	2	elintab 4890	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
4	1, 3	bitr4i 277	. 2 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥})
5	4	eqriv 2735	1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  ∩ cint 4879  Lim wlim 6267  ωcom 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-om 7713

This theorem is referenced by: (None)