Theorem bnj22 13191

Description: Technical lemma of bnj7 13196. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem).

Hypotheses

Ref	Expression
bnj22.1	|- (th <-> A.x e. A ph)
bnj22.2	|- B = {x e. A | -. ph}
bnj22.3	|- A e. _V

Assertion

Ref	Expression
bnj22	|- ((R Fr A /\ -. th) -> E.y e. B A.z e. B -. zRy)

Distinct variable groups:   x,A   y,A,z   y,B,z   y,R,z

Proof of Theorem bnj22

Step	Hyp	Ref	Expression
1		fri 3626	. 2 |- (((B e. _V /\ R Fr A) /\ (B C_ A /\ B =/= (/))) -> E.y e. B A.z e. B -. zRy)
2		bnj22.2	. . . 4 |- B = {x e. A | -. ph}
3		bnj22.3	. . . 4 |- A e. _V
4	2, 3	bnj20 12386	. . 3 |- B e. _V
5	4	biantrur 794	. 2 |- (R Fr A <-> (B e. _V /\ R Fr A))
6		bnj22.1	. . . 4 |- (th <-> A.x e. A ph)
7	6, 2	bnj19 12385	. . 3 |- (-. th <-> B =/= (/))
8	2	bnj21 12387	. . . 4 |- B C_ A
9	8	biantrur 794	. . 3 |- (B =/= (/) <-> (B C_ A /\ B =/= (/)))
10	7, 9	bitri 190	. 2 |- (-. th <-> (B C_ A /\ B =/= (/)))
11	1, 5, 10	syl2anb 504	1 |- ((R Fr A /\ -. th) -> E.y e. B A.z e. B -. zRy)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  (/)c0 2875   class class class wbr 3338   Fr wfr 3623

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  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-in 2603  df-ss 2605  df-nul 2876  df-fr 3625

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