Theorem rcfpfillem2 10564

Description: Lemma for rcfpfil 10569.

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 2331	. . . . . . . 8 |- (A (_ b <-> (A \ b) = (/))
2		eqcom 1480	. . . . . . . 8 |- ((A \ b) = (/) <-> (/) = (A \ b))
3	1, 2	bitr 173	. . . . . . 7 |- (A (_ b <-> (/) = (A \ b))
4		eqcom 1480	. . . . . . . . . . 11 |- (b = A <-> A = b)
5		eqss 2080	. . . . . . . . . . 11 |- (A = b <-> (A (_ b /\ b (_ A))
6	4, 5	bitr 173	. . . . . . . . . 10 |- (b = A <-> (A (_ b /\ b (_ A))
7		eleq1 1537	. . . . . . . . . . 11 |- (b = A -> (b e. Fin <-> A e. Fin))
8	7	biimpd 153	. . . . . . . . . 10 |- (b = A -> (b e. Fin -> A e. Fin))
9	6, 8	sylbir 201	. . . . . . . . 9 |- ((A (_ b /\ b (_ A) -> (b e. Fin -> A e. Fin))
10	9	ex 373	. . . . . . . 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 201	. . . . . 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 829	. . . 4 |- ((b (_ A /\ b e. Fin /\ (/) = (A \ b)) -> A e. Fin)
15	14	19.23aiv 1297	. . 3 |- (E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b)) -> A e. Fin)
16	15	con3i 98	. 2 |- (-. A e. Fin -> -. E.b(b (_ A /\ b e. Fin /\ (/) = (A \ b)))
17		0ex 2716	. . 3 |- (/) e. V
18		rcfpfillem1 10563	. . . 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	negbid 613	. . 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 200	1 |- (-. A e. Fin -> -. (/) e. {x | E.b(b (_ A /\ b e. Fin /\ x = (A \ b))})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  Vcvv 1814   \ cdif 2047   (_ wss 2050  (/)c0 2283  Fincfn 4373

This theorem is referenced by:  rcfpfil 10569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284

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