Theorem bnj98 13221

Description: Technical lemma of bnj109 13226. 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).

Assertion

Ref	Expression
bnj98	|- A.i e. om (suc i e. 1o -> (F` suc i) = U_y e. (F` i) pred(y, A, R))

Proof of Theorem bnj98

Step	Hyp	Ref	Expression
1		alral 2153	. 2 |- (A.i(suc i e. 1o -> (F` suc i) = U_y e. (F` i) pred(y, A, R)) -> A.i e. om (suc i e. 1o -> (F` suc i) = U_y e. (F` i) pred(y, A, R)))
2		visset 2295	. . . . . 6 |- i e. _V
3	2	sucid 3744	. . . . 5 |- i e. suc i
4		n0i 2880	. . . . 5 |- (i e. suc i -> -. suc i = (/))
5	3, 4	ax-mp 7	. . . 4 |- -. suc i = (/)
6		elsni 3066	. . . . 5 |- ({x | (x e. i \/ x e. {i})} e. {(/)} -> {x | (x e. i \/ x e. {i})} = (/))
7		df-suc 3663	. . . . . . 7 |- suc i = (i u. {i})
8		df-un 2600	. . . . . . 7 |- (i u. {i}) = {x | (x e. i \/ x e. {i})}
9	7, 8	eqtri 1908	. . . . . 6 |- suc i = {x | (x e. i \/ x e. {i})}
10		df1o2 5185	. . . . . 6 |- 1o = {(/)}
11	9, 10	eleq12i 1962	. . . . 5 |- (suc i e. 1o <-> {x | (x e. i \/ x e. {i})} e. {(/)})
12	9	eqeq1i 1891	. . . . 5 |- (suc i = (/) <-> {x | (x e. i \/ x e. {i})} = (/))
13	6, 11, 12	3imtr4i 236	. . . 4 |- (suc i e. 1o -> suc i = (/))
14	5, 13	mto 121	. . 3 |- -. suc i e. 1o
15	14	pm2.21i 93	. 2 |- (suc i e. 1o -> (F` suc i) = U_y e. (F` i) pred(y, A, R))
16	1, 15	mpg 1332	1 |- A.i e. om (suc i e. 1o -> (F` suc i) = U_y e. (F` i) pred(y, A, R))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105   u. cun 2591  (/)c0 2875  {csn 3044  U_ciun 3255  suc csuc 3659  omcom 3949  ` cfv 3998  1oc1o 5172   predsyn-bnj14 12023

This theorem is referenced by:  bnj109 13226  bnj128 13238

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-nul 2876  df-sn 3049  df-suc 3663  df-1o 5177

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