Theorem orddisj 4650

Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
orddisj	|- ( Ord A -> ( A i^i { A } ) = (/) )

Proof of Theorem orddisj

Step	Hyp	Ref	Expression
1		ordirr 4646	. 2 |- ( Ord A -> -. A e. A )
2		disjsn 3735	. 2 |- ( ( A i^i { A } ) = (/) <-> -. A e. A )
3	1, 2	sylibr 134	1 |- ( Ord A -> ( A i^i { A } ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2202   i^i cin 3200   (/)c0 3496   {csn 3673   Ord word 4465

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-in 3207  df-nul 3497  df-sn 3679

This theorem is referenced by:  orddif  4651  phplem2  7082  ennnfonelemhf1o  13097