Theorem 2wlkdlem7 26828

Description: Lemma 7 for 2wlkd 26832. (Contributed by AV, 14-Feb-2021.)

Hypotheses

Ref	Expression
2wlkd.p	|- P = <" A B C ">
2wlkd.f	|- F = <" J K ">
2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )

Assertion

Ref	Expression
2wlkdlem7	|- ( ph -> ( J e. _V /\ K e. _V ) )

Proof of Theorem 2wlkdlem7

Step	Hyp	Ref	Expression
1		2wlkd.p	. . 3 |- P = <" A B C ">
2		2wlkd.f	. . 3 |- F = <" J K ">
3		2wlkd.s	. . 3 |- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4		2wlkd.n	. . 3 |- ( ph -> ( A =/= B /\ B =/= C ) )
5		2wlkd.e	. . 3 |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6	1, 2, 3, 4, 5	2wlkdlem6 26827	. 2 |- ( ph -> ( B e. ( I ` J ) /\ B e. ( I ` K ) ) )
7		elfvex 6221	. . 3 |- ( B e. ( I ` J ) -> J e. _V )
8		elfvex 6221	. . 3 |- ( B e. ( I ` K ) -> K e. _V )
9	7, 8	anim12i 590	. 2 |- ( ( B e. ( I ` J ) /\ B e. ( I ` K ) ) -> ( J e. _V /\ K e. _V ) )
10	6, 9	syl 17	1 |- ( ph -> ( J e. _V /\ K e. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {cpr 4179   ` cfv 5888   <"cs2 13586   <"cs3 13587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by:  2wlkdlem8  26829  2trld  26834