Theorem emhgrat 10746

Description: An ordered pair with an empty second element is a hypergraph (with no blocks/edges).

Assertion

Ref	Expression
emhgrat	|- (A e. B -> <.A, (/)>. e. HypGrph)

Proof of Theorem emhgrat

Step	Hyp	Ref	Expression
1		in0 2302	. . 3 |- (A i^i (/)) = (/)
2		ral0 2362	. . 3 |- A.b e. (/) (b (_ A /\ b =/= (/))
3	1, 2	pm3.2i 285	. 2 |- ((A i^i (/)) = (/) /\ A.b e. (/) (b (_ A /\ b =/= (/)))
4		0ex 2716	. . 3 |- (/) e. V
5		eqid 1478	. . . 4 |- <.A, (/)>. = <.A, (/)>.
6	5	ishgrag 10740	. . 3 |- ((A e. B /\ (/) e. V) -> (<.A, (/)>. e. HypGrph <-> ((A i^i (/)) = (/) /\ A.b e. (/) (b (_ A /\ b =/= (/)))))
7	4, 6	mpan2 698	. 2 |- (A e. B -> (<.A, (/)>. e. HypGrph <-> ((A i^i (/)) = (/) /\ A.b e. (/) (b (_ A /\ b =/= (/)))))
8	3, 7	mpbiri 194	1 |- (A e. B -> <.A, (/)>. e. HypGrph)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  Vcvv 1814   i^i cin 2049   (_ wss 2050  (/)c0 2283  <.cop 2415  HypGrphchgra 10736

This theorem is referenced by:  0hgra 10747

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-hgra 10737

Copyright terms: Public domain