Theorem hgrablkne0 10773

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 10772	. . . 4 |- (H e. HypGrph -> B (_ (P~(1st` H) \ {(/)}))
4		blkssatm 10767	. . . 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 1694	. . 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 1713	1 |- (H e. HypGrph -> A.b e. B b =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645   \ cdif 2044   (_ wss 2047  (/)c0 2280  P~cpw 2401  {csn 2409  ` cfv 3182  1stc1st 4077  2ndc2nd 4078  HypGrphchgra 10765

This theorem is referenced by:  hgrablkcard 10774

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-9 965  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  ax-un 2866

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-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079  df-2nd 4080  df-hgra 10766

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