Theorem cplem2 4645

Description: Lemma for the Collection Principle cp 4646.

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 1452	. . 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 1452	. . 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 4644	. 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 4640	. . . 4 |- {z e. B | A.w e. B (rank` z) (_ (rank` w)} e. V
6	4, 5	iunex 3802	. . 3 |- U_x e. A {z e. B | A.w e. B (rank` z) (_ (rank` w)} e. V
7		hbiu1 2552	. . . . 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 1543	. . . 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 2182	. . . . . 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 1570	. . . . 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 610	. . . 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 1637	. . 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 1842	. 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  E.wex 956   = wceq 1099   e. wcel 1105   =/= wne 1561  A.wral 1621  {crab 1624  Vcvv 1786   i^i cin 2017   (_ wss 2018  (/)c0 2251  U_ciun 2534  ` cfv 3145  rankcrnk 4566

This theorem is referenced by:  cp 4646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-reg 4517  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-rab 1628  df-v 1787  df-sbc 1913  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-iin 2537  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161  df-rdg 3871  df-r1 4567  df-rank 4568

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