Theorem dfom5 9673

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 9671	. . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
2		vex 3468	. . . 4 ⊢ 𝑦 ∈ V
3	2	elintab 4940	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥} ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
4	1, 3	bitr4i 278	. 2 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∩ {𝑥 ∣ Lim 𝑥})
5	4	eqriv 2731	1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Syntax hints:   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2712  ∩ cint 4928  Lim wlim 6366  ωcom 7870

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 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738  ax-inf2 9664

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-int 4929  df-br 5126  df-opab 5188  df-tr 5242  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7871

This theorem is referenced by: (None)