Theorem inton 4428

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 4427	. 2 |- (/) e. On
2		int0el 3904	. 2 |- ( (/) e. On -> |^| On = (/) )
3	1, 2	ax-mp 5	1 |- |^| On = (/)

Syntax hints:   = wceq 1364   e. wcel 2167   (/)c0 3450   |^|cint 3874   Oncon0 4398

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-uni 3840  df-int 3875  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by: (None)