Theorem inton 5699

Description: The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)

Assertion

Ref	Expression
inton	⊢ ∩ On = ∅

Proof of Theorem inton

Step	Hyp	Ref	Expression
1		0elon 5695	. 2 ⊢ ∅ ∈ On
2		int0el 4443	. 2 ⊢ (∅ ∈ On → ∩ On = ∅)
3	1, 2	ax-mp 5	1 ⊢ ∩ On = ∅

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ∩ cint 4410  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-uni 4373  df-int 4411  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by: (None)