Theorem inton 4484

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 4483	. 2 |- (/) e. On
2		int0el 3953	. 2 |- ( (/) e. On -> |^| On = (/) )
3	1, 2	ax-mp 5	1 |- |^| On = (/)

Syntax hints:   = wceq 1395   e. wcel 2200   (/)c0 3491   |^|cint 3923   Oncon0 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4210

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-uni 3889  df-int 3924  df-tr 4183  df-iord 4457  df-on 4459

This theorem is referenced by: (None)