Theorem emhgrat 10775

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 2298	. . 3 |- (A i^i (/)) = (/)
2		ral0 2358	. . 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 2711	. . 3 |- (/) e. V
5		eqid 1475	. . . 4 |- <.A, (/)>. = <.A, (/)>.
6	5	ishgrag 10769	. . 3 |- ((A e. B /\ (/) e. V) -> (<.A, (/)>. e. HypGrph <-> ((A i^i (/)) = (/) /\ A.b e. (/) (b (_ A /\ b =/= (/)))))
7	4, 6	mpan2 696	. 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 956   e. wcel 958   =/= wne 1585  A.wral 1645  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  <.cop 2411  HypGrphchgra 10765

This theorem is referenced by:  0hgra 10776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-hgra 10766

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