Theorem dfom5 9618

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). Theorem 1.23 of [Schloeder] p. 4. (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 9616	. . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
2		vex 3467	. . . 4 ⊢ 𝑦 ∈ V
3	2	elintab 4928	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
4	1, 3	bitr4i 281	. 2 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥})
5	4	eqriv 2766	1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Syntax hints:   → wi 4  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  ∩ cint 4916  Lim wlim 6362  ωcom 7861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-inf2 9609

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-om 7862

This theorem is referenced by: (None)