Theorem cflem 4917

Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set A.

Assertion

Ref	Expression
cflem	|- (A e. B -> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))

Distinct variable group:   x,y,z,w,A

Proof of Theorem cflem

Step	Hyp	Ref	Expression
1		ssid 2083	. . 3 |- A (_ A
2		ssid 2083	. . . . 5 |- z (_ z
3		sseq2 2086	. . . . . 6 |- (w = z -> (z (_ w <-> z (_ z))
4	3	rcla4ev 1880	. . . . 5 |- ((z e. A /\ z (_ z) -> E.w e. A z (_ w)
5	2, 4	mpan2 698	. . . 4 |- (z e. A -> E.w e. A z (_ w)
6	5	rgen 1701	. . 3 |- A.z e. A E.w e. A z (_ w
7		sseq1 2085	. . . . 5 |- (y = A -> (y (_ A <-> A (_ A))
8		rexeq1 1790	. . . . . 6 |- (y = A -> (E.w e. y z (_ w <-> E.w e. A z (_ w))
9	8	ralbidv 1666	. . . . 5 |- (y = A -> (A.z e. A E.w e. y z (_ w <-> A.z e. A E.w e. A z (_ w))
10	7, 9	anbi12d 630	. . . 4 |- (y = A -> ((y (_ A /\ A.z e. A E.w e. y z (_ w) <-> (A (_ A /\ A.z e. A E.w e. A z (_ w)))
11	10	cla4egv 1866	. . 3 |- (A e. B -> ((A (_ A /\ A.z e. A E.w e. A z (_ w) -> E.y(y (_ A /\ A.z e. A E.w e. y z (_ w)))
12	1, 6, 11	mp2ani 702	. 2 |- (A e. B -> E.y(y (_ A /\ A.z e. A E.w e. y z (_ w))
13		19.41v 1307	. . . . 5 |- (E.x(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (E.x x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
14		fvex 3738	. . . . . 6 |- (card` y) e. V
15	14	isseti 1818	. . . . 5 |- E.x x = (card` y)
16	13, 15	mpbiran 730	. . . 4 |- (E.x(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (y (_ A /\ A.z e. A E.w e. y z (_ w))
17	16	exbii 1053	. . 3 |- (E.yE.x(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y(y (_ A /\ A.z e. A E.w e. y z (_ w))
18		excom 1048	. . 3 |- (E.yE.x(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
19	17, 18	bitr3 175	. 2 |- (E.y(y (_ A /\ A.z e. A E.w e. y z (_ w) <-> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
20	12, 19	sylib 198	1 |- (A e. B -> E.xE.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  A.wral 1648  E.wrex 1649   (_ wss 2050  ` cfv 3188  cardccrd 4823

This theorem is referenced by:  cfval 4918  cffnon 4919

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-13 971  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-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508  df-fv 3204

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