Theorem dfom5 9571

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 9569	. . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
2		vex 3433	. . . 4 ⊢ 𝑦 ∈ V
3	2	elintab 4901	. . 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 2733	1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2714  ∩ cint 4889  Lim wlim 6324  ωcom 7817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-tr 5193  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-om 7818

This theorem is referenced by: (None)