Theorem cplem2 5851

Description: -Lemma for the Collection Principle cp 5852.

Hypothesis

Ref	Expression
cplem2.1	|- A e. _V

Assertion

Ref	Expression
cplem2	|- E.yA.x e. A (B =/= (/) -> (B i^i y) =/= (/))

Distinct variable groups:   x,y,A   y,B

Proof of Theorem cplem2

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- {z e. B | A.w e. B (rank` z) C_ (rank` w)} = {z e. B | A.w e. B (rank` z) C_ (rank` w)}
2		eqid 1884	. . 3 |- U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}
3	1, 2	cplem1 5850	. 2 |- A.x e. A (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}) =/= (/))
4		cplem2.1	. . . 4 |- A e. _V
5		scottex 5846	. . . 4 |- {z e. B | A.w e. B (rank` z) C_ (rank` w)} e. _V
6	4, 5	iunex 4839	. . 3 |- U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} e. _V
7		hbiu1 3281	. . . . 5 |- (y e. U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> A.x y e. U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)})
8	7	hbeleq 1997	. . . 4 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> A.x y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)})
9		ineq2 2790	. . . . . 6 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> (B i^i y) = (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}))
10	9	neeq1d 2028	. . . . 5 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> ((B i^i y) =/= (/) <-> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}) =/= (/)))
11	10	imbi2d 674	. . . 4 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> ((B =/= (/) -> (B i^i y) =/= (/)) <-> (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}) =/= (/))))
12	8, 11	ralbid 2121	. . 3 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)} -> (A.x e. A (B =/= (/) -> (B i^i y) =/= (/)) <-> A.x e. A (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}) =/= (/))))
13	6, 12	cla4ev 2371	. 2 |- (A.x e. A (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) C_ (rank` w)}) =/= (/)) -> E.yA.x e. A (B =/= (/) -> (B i^i y) =/= (/)))
14	3, 13	ax-mp 7	1 |- E.yA.x e. A (B =/= (/) -> (B i^i y) =/= (/))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  {crab 2108  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  U_ciun 3255  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  cp 5852

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-13 1311  ax-14 1312  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  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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