Theorem blkssatm 10738

Description: Two ways of saying "the blocks in a hypergraph are each non-empty subsets of the set of atoms in the hypergraph."

Assertion

Ref	Expression
blkssatm	|- (B (_ (P~A \ {(/)}) <-> A.b e. B (b (_ A /\ b =/= (/)))

Distinct variable groups:   A,b   B,b

Proof of Theorem blkssatm

Step	Hyp	Ref	Expression
1		dfss3 2062	. 2 |- (B (_ (P~A \ {(/)}) <-> A.b e. B b e. (P~A \ {(/)}))
2		eldifsn 2466	. . . 4 |- (b e. (P~A \ {(/)}) <-> (b e. P~A /\ b =/= (/)))
3		visset 1816	. . . . . 6 |- b e. V
4	3	elpw 2408	. . . . 5 |- (b e. P~A <-> b (_ A)
5	4	anbi1i 483	. . . 4 |- ((b e. P~A /\ b =/= (/)) <-> (b (_ A /\ b =/= (/)))
6	2, 5	bitr 173	. . 3 |- (b e. (P~A \ {(/)}) <-> (b (_ A /\ b =/= (/)))
7	6	ralbii 1670	. 2 |- (A.b e. B b e. (P~A \ {(/)}) <-> A.b e. B (b (_ A /\ b =/= (/)))
8	1, 7	bitr 173	1 |- (B (_ (P~A \ {(/)}) <-> A.b e. B (b (_ A /\ b =/= (/)))

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 960   =/= wne 1588  A.wral 1648   \ cdif 2047   (_ wss 2050  (/)c0 2283  P~cpw 2405  {csn 2413

This theorem is referenced by:  ishgrag 10740  hgrablkne0 10744

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  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-pw 2406  df-sn 2416  df-pr 2417

Copyright terms: Public domain