Theorem dfpw12 4301

Description: Alternate expression for unit power classes. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
dfpw12	|- ~P1 A = ( SIk (A X.k A)"k_V)

Proof of Theorem dfpw12

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw1 4144	. . 3 |- (x e. ~P1 A <-> E.y e. A x = {y})
2		vex 2862	. . . . 5 |- x e. _V
3	2	elimakv 4260	. . . 4 |- (x e. ( SIk (A X.k A)"k_V) <-> E.z<<z, x>> e. SIk (A X.k A))
4		vex 2862	. . . . . . 7 |- z e. _V
5		opkelsikg 4264	. . . . . . 7 |- ((z e. _V /\ x e. _V) -> (<<z, x>> e. SIk (A X.k A) <-> E.wE.y(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A))))
6	4, 2, 5	mp2an 653	. . . . . 6 |- (<<z, x>> e. SIk (A X.k A) <-> E.wE.y(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)))
7	6	exbii 1582	. . . . 5 |- (E.z<<z, x>> e. SIk (A X.k A) <-> E.zE.wE.y(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)))
8		exrot3 1744	. . . . 5 |- (E.yE.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> E.zE.wE.y(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)))
9	7, 8	bitr4i 243	. . . 4 |- (E.z<<z, x>> e. SIk (A X.k A) <-> E.yE.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)))
10		df-3an 936	. . . . . . . . 9 |- ((z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> ((z = {w} /\ x = {y}) /\ <<w, y>> e. (A X.k A)))
11		vex 2862	. . . . . . . . . . 11 |- w e. _V
12		vex 2862	. . . . . . . . . . 11 |- y e. _V
13	11, 12	opkelxpk 4248	. . . . . . . . . 10 |- (<<w, y>> e. (A X.k A) <-> (w e. A /\ y e. A))
14	13	anbi2i 675	. . . . . . . . 9 |- (((z = {w} /\ x = {y}) /\ <<w, y>> e. (A X.k A)) <-> ((z = {w} /\ x = {y}) /\ (w e. A /\ y e. A)))
15		an4 797	. . . . . . . . 9 |- (((z = {w} /\ x = {y}) /\ (w e. A /\ y e. A)) <-> ((z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)))
16	10, 14, 15	3bitri 262	. . . . . . . 8 |- ((z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> ((z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)))
17	16	2exbii 1583	. . . . . . 7 |- (E.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> E.zE.w((z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)))
18		19.41vv 1902	. . . . . . 7 |- (E.zE.w((z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)) <-> (E.zE.w(z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)))
19		sneq 3744	. . . . . . . . . . . 12 |- (w = y -> {w} = {y})
20		eqeq12 2365	. . . . . . . . . . . 12 |- ((z = x /\ {w} = {y}) -> (z = {w} <-> x = {y}))
21	19, 20	sylan2 460	. . . . . . . . . . 11 |- ((z = x /\ w = y) -> (z = {w} <-> x = {y}))
22		eleq1 2413	. . . . . . . . . . . 12 |- (w = y -> (w e. A <-> y e. A))
23	22	adantl 452	. . . . . . . . . . 11 |- ((z = x /\ w = y) -> (w e. A <-> y e. A))
24	21, 23	anbi12d 691	. . . . . . . . . 10 |- ((z = x /\ w = y) -> ((z = {w} /\ w e. A) <-> (x = {y} /\ y e. A)))
25	2, 12, 24	spc2ev 2947	. . . . . . . . 9 |- ((x = {y} /\ y e. A) -> E.zE.w(z = {w} /\ w e. A))
26	25	pm4.71ri 614	. . . . . . . 8 |- ((x = {y} /\ y e. A) <-> (E.zE.w(z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)))
27		ancom 437	. . . . . . . 8 |- ((x = {y} /\ y e. A) <-> (y e. A /\ x = {y}))
28	26, 27	bitr3i 242	. . . . . . 7 |- ((E.zE.w(z = {w} /\ w e. A) /\ (x = {y} /\ y e. A)) <-> (y e. A /\ x = {y}))
29	17, 18, 28	3bitri 262	. . . . . 6 |- (E.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> (y e. A /\ x = {y}))
30	29	exbii 1582	. . . . 5 |- (E.yE.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> E.y(y e. A /\ x = {y}))
31		df-rex 2620	. . . . 5 |- (E.y e. A x = {y} <-> E.y(y e. A /\ x = {y}))
32	30, 31	bitr4i 243	. . . 4 |- (E.yE.zE.w(z = {w} /\ x = {y} /\ <<w, y>> e. (A X.k A)) <-> E.y e. A x = {y})
33	3, 9, 32	3bitri 262	. . 3 |- (x e. ( SIk (A X.k A)"k_V) <-> E.y e. A x = {y})
34	1, 33	bitr4i 243	. 2 |- (x e. ~P1 A <-> x e. ( SIk (A X.k A)"k_V))
35	34	eqriv 2350	1 |- ~P1 A = ( SIk (A X.k A)"k_V)

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2615  _Vcvv 2859  {csn 3737  <<copk 4057  ~P₁ cpw1 4135   X._k cxpk 4174  "_kcimak 4179   SI_k csik 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-imak 4189  df-sik 4192

This theorem is referenced by:  pw1exg  4302