Theorem hgrablkne0 10717

Description: The empty set cannot be a block in a hypergraph.

Hypothesis

Ref	Expression
hgrablkne0.1	|- B = (2nd` H)

Assertion

Ref	Expression
hgrablkne0	|- (H e. HypGrph -> A.b e. B b =/= (/))

Distinct variable groups:   H,b   B,b

Proof of Theorem hgrablkne0

Step	Hyp	Ref	Expression
1		eqid 1475	. . . . 5 |- (1st` H) = (1st` H)
2		hgrablkne0.1	. . . . 5 |- B = (2nd` H)
3	1, 2	hgrablkconn 10716	. . . 4 |- (H e. HypGrph -> B (_ (P~(1st` H) \ {(/)}))
4		blkssatm 10711	. . . 4 |- (B (_ (P~(1st` H) \ {(/)}) <-> A.b e. B (b (_ (1st` H) /\ b =/= (/)))
5	3, 4	sylib 198	. . 3 |- (H e. HypGrph -> A.b e. B (b (_ (1st` H) /\ b =/= (/)))
6		ra4 1693	. . 3 |- (A.b e. B (b (_ (1st` H) /\ b =/= (/)) -> (b e. B -> (b (_ (1st` H) /\ b =/= (/))))
7		pm3.27 323	. . . 4 |- ((b (_ (1st` H) /\ b =/= (/)) -> b =/= (/))
8	7	imim2i 17	. . 3 |- ((b e. B -> (b (_ (1st` H) /\ b =/= (/))) -> (b e. B -> b =/= (/)))
9	5, 6, 8	3syl 20	. 2 |- (H e. HypGrph -> (b e. B -> b =/= (/)))
10	9	r19.21aiv 1712	1 |- (H e. HypGrph -> A.b e. B b =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957   =/= wne 1584  A.wral 1644   \ cdif 2042   (_ wss 2045  (/)c0 2278  P~cpw 2399  {csn 2407  ` cfv 3179  1stc1st 4074  2ndc2nd 4075  HypGrphchgra 10709

This theorem is referenced by:  hgrablkcard 10718

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fv 3195  df-1st 4076  df-2nd 4077  df-hgra 10710

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