Theorem rcfpfillem2 10672

Description: Lemma for rcfpfil 10677.

Assertion

Ref	Expression
rcfpfillem2	|- (-. A e. Fin -> -. (/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))})

Distinct variable group:   A,b,x

Proof of Theorem rcfpfillem2

Step	Hyp	Ref	Expression
1		ssdif0 2379	. . . . . . . 8 |- (A (_ b <-> (A \ b) = (/))
2		eqcom 1524	. . . . . . . 8 |- ((A \ b) = (/) <-> (/) = (A \ b))
3	1, 2	bitri 180	. . . . . . 7 |- (A (_ b <-> (/) = (A \ b))
4		eqcom 1524	. . . . . . . . . . 11 |- (b = A <-> A = b)
5		eqss 2128	. . . . . . . . . . 11 |- (A = b <-> (A (_ b /\ b (_ A))
6	4, 5	bitri 180	. . . . . . . . . 10 |- (b = A <-> (A (_ b /\ b (_ A))
7		eleq1 1581	. . . . . . . . . . 11 |- (b = A -> (b e. Fin <-> A e. Fin))
8	7	biimpd 160	. . . . . . . . . 10 |- (b = A -> (b e. Fin -> A e. Fin))
9	6, 8	sylbir 208	. . . . . . . . 9 |- ((A (_ b /\ b (_ A) -> (b e. Fin -> A e. Fin))
10	9	ex 380	. . . . . . . 8 |- (A (_ b -> (b (_ A -> (b e. Fin -> A e. Fin)))
11	10	com23 32	. . . . . . 7 |- (A (_ b -> (b e. Fin -> (b (_ A -> A e. Fin)))
12	3, 11	sylbir 208	. . . . . 6 |- ((/) = (A \ b) -> (b e. Fin -> (b (_ A -> A e. Fin)))
13	12	com13 33	. . . . 5 |- (b (_ A -> (b e. Fin -> ((/) = (A \ b) -> A e. Fin)))
14	13	3imp 839	. . . 4 |- ((b (_ A /\ b e. Fin /\ (/) = (A \ b)) -> A e. Fin)
15	14	19.23aiv 1337	. . 3 |- (E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b)) -> A e. Fin)
16	15	con3i 104	. 2 |- (-. A e. Fin -> -. E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b)))
17		0ex 2766	. . 3 |- (/) e. V
18		rcfpfillem1 10671	. . . 4 |- ((/) e. V -> ((/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} <-> E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b))))
19	18	notbid 622	. . 3 |- ((/) e. V -> (-. (/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} <-> -. E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b))))
20	17, 19	ax-mp 7	. 2 |- (-. (/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))} <-> -. E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b)))
21	16, 20	sylibr 207	1 |- (-. A e. Fin -> -. (/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230   /\ w3a 787   = wceq 997   e. wcel 999  E.wex 1021  {cab 1509  Vcvv 1858   \ cdif 2095   (_ wss 2098  (/)c0 2331  Fincfn 4428

This theorem is referenced by:  rcfpfil 10677

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-nul 2765

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-in 2102  df-ss 2104  df-nul 2332

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