Metamath Proof Explorer


Theorem bnj1177

Description: Technical lemma for bnj69 . 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). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1177.2 ψ X B y B y R X
bnj1177.3 C = trCl X A R B
bnj1177.9 φ ψ R FrSe A
bnj1177.13 φ ψ B A
bnj1177.17 φ ψ X A
Assertion bnj1177 φ ψ R Fr A C A C C V

Proof

Step Hyp Ref Expression
1 bnj1177.2 ψ X B y B y R X
2 bnj1177.3 C = trCl X A R B
3 bnj1177.9 φ ψ R FrSe A
4 bnj1177.13 φ ψ B A
5 bnj1177.17 φ ψ X A
6 df-bnj15 R FrSe A R Fr A R Se A
7 6 simplbi R FrSe A R Fr A
8 3 7 syl φ ψ R Fr A
9 bnj1147 trCl X A R A
10 ssinss1 trCl X A R A trCl X A R B A
11 9 10 ax-mp trCl X A R B A
12 2 11 eqsstri C A
13 12 a1i φ ψ C A
14 bnj906 R FrSe A X A pred X A R trCl X A R
15 3 5 14 syl2anc φ ψ pred X A R trCl X A R
16 15 ssrind φ ψ pred X A R B trCl X A R B
17 1 simp2bi ψ y B
18 17 adantl φ ψ y B
19 4 18 sseldd φ ψ y A
20 1 simp3bi ψ y R X
21 20 adantl φ ψ y R X
22 bnj1152 y pred X A R y A y R X
23 19 21 22 sylanbrc φ ψ y pred X A R
24 23 18 elind φ ψ y pred X A R B
25 16 24 sseldd φ ψ y trCl X A R B
26 25 ne0d φ ψ trCl X A R B
27 2 neeq1i C trCl X A R B
28 26 27 sylibr φ ψ C
29 bnj893 R FrSe A X A trCl X A R V
30 3 5 29 syl2anc φ ψ trCl X A R V
31 inex1g trCl X A R V trCl X A R B V
32 2 31 eqeltrid trCl X A R V C V
33 30 32 syl φ ψ C V
34 8 13 28 33 bnj951 φ ψ R Fr A C A C C V