Theorem inton 4148

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 4147	. 2 |- (/) e. On
2		int0el 3666	. 2 |- ( (/) e. On -> |^| On = (/) )
3	1, 2	ax-mp 7	1 |- |^| On = (/)

Syntax hints:   = wceq 1284   e. wcel 1433   (/)c0 3251   |^|cint 3636   Oncon0 4118

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-uni 3602  df-int 3637  df-tr 3876  df-iord 4121  df-on 4123

This theorem is referenced by: (None)