Theorem dfom5 9615

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 9613	. . 3 ⊢ (𝑦 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝑦 ∈ 𝑥))
2		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
3	2	elintab 4924	. . 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 2760	1 ⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}

Syntax hints:   → wi 4  ∀wal 1568   = wceq 1570   ∈ wcel 2143  {cab 2741  ∩ cint 4912  Lim wlim 6361  ωcom 7858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732  ax-inf2 9606

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-om 7859

This theorem is referenced by: (None)