Theorem inton 4371

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 4370	. 2 |- (/) e. On
2		int0el 3854	. 2 |- ( (/) e. On -> |^| On = (/) )
3	1, 2	ax-mp 5	1 |- |^| On = (/)

Syntax hints:   = wceq 1343   e. wcel 2136   (/)c0 3409   |^|cint 3824   Oncon0 4341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-uni 3790  df-int 3825  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by: (None)