Theorem cplem2 4721

Description: Lemma for the Collection Principle cp 4722.

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 1475	. . 3 |- {z e. B | A.w e. B (rank` z) (_ (rank` w)} = {z e. B | A.w e. B (rank` z) (_ (rank` w)}
2		eqid 1475	. . 3 |- U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)}
3	1, 2	cplem1 4720	. 2 |- A.x e. A (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)}) =/= (/))
4		cplem2.1	. . . 4 |- A e. V
5		scottex 4716	. . . 4 |- {z e. B | A.w e. B (rank` z) (_ (rank` w)} e. V
6	4, 5	iunex 3863	. . 3 |- U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} e. V
7		hbiu1 2584	. . . . 5 |- (y e. U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} -> A.x y e. U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)})
8	7	hbeleq 1567	. . . 4 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} -> A.x y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)})
9		ineq2 2211	. . . . . 6 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} -> (B i^i y) = (B i^i U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)}))
10	9	neeq1d 1594	. . . . 5 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} -> ((B i^i y) =/= (/) <-> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)}) =/= (/)))
11	10	imbi2d 612	. . . 4 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} -> ((B =/= (/) -> (B i^i y) =/= (/)) <-> (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)}) =/= (/))))
12	8, 11	ralbid 1661	. . 3 |- (y = U_x e. A {z e. B | A.w e. B (rank` z) (_ (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) (_ (rank` w)}) =/= (/))))
13	6, 12	cla4ev 1869	. 2 |- (A.x e. A (B =/= (/) -> (B i^i U_x e. A {z e. B | A.w e. B (rank` z) (_ (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 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  {crab 1648  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  U_ciun 2566  ` cfv 3182  rankcrnk 4642

This theorem is referenced by:  cp 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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