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Theorem en3lplem2 7922
Description: Lemma for en3lp 7923. (Contributed by Alan Sare, 28-Oct-2011.)
Assertion
Ref Expression
en3lplem2 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x i^i { A , B , C } ) =/= (/) ) )
Distinct variable groups:   x, A   x, B   x, C
Proof of Theorem en3lplem2
StepHypRef Expression
1 en3lplem1 7921 . . . . 5 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = A -> ( x i^i { A , B , C } ) =/= (/) ) )
2 en3lplem1 7921 . . . . . . . 8 |- ( ( B e. C /\ C e. A /\ A e. B ) -> ( x = B -> ( x i^i { B , C , A } ) =/= (/) ) )
323comr 1196 . . . . . . 7 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = B -> ( x i^i { B , C , A } ) =/= (/) ) )
43a1d 25 . . . . . 6 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x = B -> ( x i^i { B , C , A } ) =/= (/) ) ) )
5 tprot 4068 . . . . . . . . 9 |- { A , B , C } = { B , C , A }
65ineq2i 3647 . . . . . . . 8 |- ( x i^i { A , B , C } ) = ( x i^i { B , C , A } )
76neeq1i 2733 . . . . . . 7 |- ( ( x i^i { A , B , C } ) =/= (/) <-> ( x i^i { B , C , A } ) =/= (/) )
87bicomi 202 . . . . . 6 |- ( ( x i^i { B , C , A } ) =/= (/) <-> ( x i^i { A , B , C } ) =/= (/) )
94, 8syl8ib 231 . . . . 5 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x = B -> ( x i^i { A , B , C } ) =/= (/) ) ) )
10 jao 512 . . . . 5 |- ( ( x = A -> ( x i^i { A , B , C } ) =/= (/) ) -> ( ( x = B -> ( x i^i { A , B , C } ) =/= (/) ) -> ( ( x = A \/ x = B ) -> ( x i^i { A , B , C } ) =/= (/) ) ) )
111, 9, 10sylsyld 56 . . . 4 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( ( x = A \/ x = B ) -> ( x i^i { A , B , C } ) =/= (/) ) ) )
1211imp 429 . . 3 |- ( ( ( A e. B /\ B e. C /\ C e. A ) /\ x e. { A , B , C } ) -> ( ( x = A \/ x = B ) -> ( x i^i { A , B , C } ) =/= (/) ) )
13 en3lplem1 7921 . . . . . . 7 |- ( ( C e. A /\ A e. B /\ B e. C ) -> ( x = C -> ( x i^i { C , A , B } ) =/= (/) ) )
14133coml 1195 . . . . . 6 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x = C -> ( x i^i { C , A , B } ) =/= (/) ) )
1514a1d 25 . . . . 5 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x = C -> ( x i^i { C , A , B } ) =/= (/) ) ) )
16 tprot 4068 . . . . . . 7 |- { C , A , B } = { A , B , C }
1716ineq2i 3647 . . . . . 6 |- ( x i^i { C , A , B } ) = ( x i^i { A , B , C } )
1817neeq1i 2733 . . . . 5 |- ( ( x i^i { C , A , B } ) =/= (/) <-> ( x i^i { A , B , C } ) =/= (/) )
1915, 18syl8ib 231 . . . 4 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x = C -> ( x i^i { A , B , C } ) =/= (/) ) ) )
2019imp 429 . . 3 |- ( ( ( A e. B /\ B e. C /\ C e. A ) /\ x e. { A , B , C } ) -> ( x = C -> ( x i^i { A , B , C } ) =/= (/) ) )
21 idd 24 . . . . . . 7 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> x e. { A , B , C } ) )
22 dftp2 4020 . . . . . . . 8 |- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
2322eleq2i 2529 . . . . . . 7 |- ( x e. { A , B , C } <-> x e. { x | ( x = A \/ x = B \/ x = C ) } )
2421, 23syl6ib 226 . . . . . 6 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> x e. { x | ( x = A \/ x = B \/ x = C ) } ) )
25 abid 2438 . . . . . 6 |- ( x e. { x | ( x = A \/ x = B \/ x = C ) } <-> ( x = A \/ x = B \/ x = C ) )
2624, 25syl6ib 226 . . . . 5 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x = A \/ x = B \/ x = C ) ) )
27 df-3or 966 . . . . 5 |- ( ( x = A \/ x = B \/ x = C ) <-> ( ( x = A \/ x = B ) \/ x = C ) )
2826, 27syl6ib 226 . . . 4 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( ( x = A \/ x = B ) \/ x = C ) ) )
2928imp 429 . . 3 |- ( ( ( A e. B /\ B e. C /\ C e. A ) /\ x e. { A , B , C } ) -> ( ( x = A \/ x = B ) \/ x = C ) )
3012, 20, 29mpjaod 381 . 2 |- ( ( ( A e. B /\ B e. C /\ C e. A ) /\ x e. { A , B , C } ) -> ( x i^i { A , B , C } ) =/= (/) )
3130ex 434 1 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x i^i { A , B , C } ) =/= (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   \/ w3o 964   /\ w3a 965   = wceq 1370   e. wcel 1758   {cab 2436   =/= wne 2644   i^i cin 3425   (/)c0 3735   {ctp 3979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-nul 3736  df-sn 3976  df-pr 3978  df-tp 3980
This theorem is referenced by:  en3lp  7923
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