Theorem hgralem 16292

Description: Lemma for various hypergraph theorems.

Hypotheses

Ref	Expression
hgralem.1	|- A = (1st` H)
hgralem.2	|- B = (2nd` H)

Assertion

Ref	Expression
hgralem	|- (H e. HypGrph -> ((A i^i B) = (/) /\ B C_ (~PA \ {(/)})))

Proof of Theorem hgralem

Step	Hyp	Ref	Expression
1		df-hgra 16288	. . 3 |- HypGrph = {<.x, y>. | ((x i^i y) = (/) /\ y C_ (~Px \ {(/)}))}
2	1	eleq2i 1961	. 2 |- (H e. HypGrph <-> H e. {<.x, y>. | ((x i^i y) = (/) /\ y C_ (~Px \ {(/)}))})
3		hgralem.1	. . . . . . 7 |- A = (1st` H)
4	3	eqeq2i 1894	. . . . . 6 |- (x = A <-> x = (1st` H))
5		ineq1 2789	. . . . . 6 |- (x = A -> (x i^i y) = (A i^i y))
6	4, 5	sylbir 218	. . . . 5 |- (x = (1st` H) -> (x i^i y) = (A i^i y))
7	6	eqeq1d 1892	. . . 4 |- (x = (1st` H) -> ((x i^i y) = (/) <-> (A i^i y) = (/)))
8		pweq 3036	. . . . . . 7 |- (x = A -> ~Px = ~PA)
9	8	difeq1d 2725	. . . . . 6 |- (x = A -> (~Px \ {(/)}) = (~PA \ {(/)}))
10	9	sseq2d 2645	. . . . 5 |- (x = A -> (y C_ (~Px \ {(/)}) <-> y C_ (~PA \ {(/)})))
11	4, 10	sylbir 218	. . . 4 |- (x = (1st` H) -> (y C_ (~Px \ {(/)}) <-> y C_ (~PA \ {(/)})))
12	7, 11	anbi12d 690	. . 3 |- (x = (1st` H) -> (((x i^i y) = (/) /\ y C_ (~Px \ {(/)})) <-> ((A i^i y) = (/) /\ y C_ (~PA \ {(/)}))))
13		hgralem.2	. . . . . . 7 |- B = (2nd` H)
14	13	eqeq2i 1894	. . . . . 6 |- (y = B <-> y = (2nd` H))
15		ineq2 2790	. . . . . 6 |- (y = B -> (A i^i y) = (A i^i B))
16	14, 15	sylbir 218	. . . . 5 |- (y = (2nd` H) -> (A i^i y) = (A i^i B))
17	16	eqeq1d 1892	. . . 4 |- (y = (2nd` H) -> ((A i^i y) = (/) <-> (A i^i B) = (/)))
18		sseq1 2637	. . . . 5 |- (y = B -> (y C_ (~PA \ {(/)}) <-> B C_ (~PA \ {(/)})))
19	14, 18	sylbir 218	. . . 4 |- (y = (2nd` H) -> (y C_ (~PA \ {(/)}) <-> B C_ (~PA \ {(/)})))
20	17, 19	anbi12d 690	. . 3 |- (y = (2nd` H) -> (((A i^i y) = (/) /\ y C_ (~PA \ {(/)})) <-> ((A i^i B) = (/) /\ B C_ (~PA \ {(/)}))))
21	12, 20	elopabi 5059	. 2 |- (H e. {<.x, y>. | ((x i^i y) = (/) /\ y C_ (~Px \ {(/)}))} -> ((A i^i B) = (/) /\ B C_ (~PA \ {(/)})))
22	2, 21	sylbi 216	1 |- (H e. HypGrph -> ((A i^i B) = (/) /\ B C_ (~PA \ {(/)})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  {copab 3395  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  HypGrphchgra 16287

This theorem is referenced by:  hgradj 16293  hgrablkconn 16294

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021  df-hgra 16288

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