Theorem inton 6327

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 6323	. 2 ⊢ ∅ ∈ On
2		int0el 4913	. 2 ⊢ (∅ ∈ On → ∩ On = ∅)
3	1, 2	ax-mp 5	1 ⊢ ∩ On = ∅

Syntax hints:   = wceq 1537   ∈ wcel 2101  ∅c0 4259  ∩ cint 4882  Oncon0 6270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-11 2149  ax-ext 2704  ax-nul 5233

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-int 4883  df-br 5078  df-tr 5195  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-ord 6273  df-on 6274

This theorem is referenced by:  bday0s  34050