Theorem pwtp 3885

Description: The power set of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
pwtp	|- ~P{A, B, C} = (({(/), {A}} u. {{B}, {A, B}}) u. ({{C}, {A, C}} u. {{B, C}, {A, B, C}}))

Proof of Theorem pwtp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 |- x e. _V
2	1	elpw 3729	. . 3 |- (x e. ~P{A, B, C} <-> x C_ {A, B, C})
3		elun 3221	. . . . . 6 |- (x e. ({(/), {A}} u. {{B}, {A, B}}) <-> (x e. {(/), {A}} \/ x e. {{B}, {A, B}}))
4	1	elpr 3752	. . . . . . 7 |- (x e. {(/), {A}} <-> (x = (/) \/ x = {A}))
5	1	elpr 3752	. . . . . . 7 |- (x e. {{B}, {A, B}} <-> (x = {B} \/ x = {A, B}))
6	4, 5	orbi12i 507	. . . . . 6 |- ((x e. {(/), {A}} \/ x e. {{B}, {A, B}}) <-> ((x = (/) \/ x = {A}) \/ (x = {B} \/ x = {A, B})))
7	3, 6	bitri 240	. . . . 5 |- (x e. ({(/), {A}} u. {{B}, {A, B}}) <-> ((x = (/) \/ x = {A}) \/ (x = {B} \/ x = {A, B})))
8		elun 3221	. . . . . 6 |- (x e. ({{C}, {A, C}} u. {{B, C}, {A, B, C}}) <-> (x e. {{C}, {A, C}} \/ x e. {{B, C}, {A, B, C}}))
9	1	elpr 3752	. . . . . . 7 |- (x e. {{C}, {A, C}} <-> (x = {C} \/ x = {A, C}))
10	1	elpr 3752	. . . . . . 7 |- (x e. {{B, C}, {A, B, C}} <-> (x = {B, C} \/ x = {A, B, C}))
11	9, 10	orbi12i 507	. . . . . 6 |- ((x e. {{C}, {A, C}} \/ x e. {{B, C}, {A, B, C}}) <-> ((x = {C} \/ x = {A, C}) \/ (x = {B, C} \/ x = {A, B, C})))
12	8, 11	bitri 240	. . . . 5 |- (x e. ({{C}, {A, C}} u. {{B, C}, {A, B, C}}) <-> ((x = {C} \/ x = {A, C}) \/ (x = {B, C} \/ x = {A, B, C})))
13	7, 12	orbi12i 507	. . . 4 |- ((x e. ({(/), {A}} u. {{B}, {A, B}}) \/ x e. ({{C}, {A, C}} u. {{B, C}, {A, B, C}})) <-> (((x = (/) \/ x = {A}) \/ (x = {B} \/ x = {A, B})) \/ ((x = {C} \/ x = {A, C}) \/ (x = {B, C} \/ x = {A, B, C}))))
14		elun 3221	. . . 4 |- (x e. (({(/), {A}} u. {{B}, {A, B}}) u. ({{C}, {A, C}} u. {{B, C}, {A, B, C}})) <-> (x e. ({(/), {A}} u. {{B}, {A, B}}) \/ x e. ({{C}, {A, C}} u. {{B, C}, {A, B, C}})))
15		sstp 3871	. . . 4 |- (x C_ {A, B, C} <-> (((x = (/) \/ x = {A}) \/ (x = {B} \/ x = {A, B})) \/ ((x = {C} \/ x = {A, C}) \/ (x = {B, C} \/ x = {A, B, C}))))
16	13, 14, 15	3bitr4ri 269	. . 3 |- (x C_ {A, B, C} <-> x e. (({(/), {A}} u. {{B}, {A, B}}) u. ({{C}, {A, C}} u. {{B, C}, {A, B, C}})))
17	2, 16	bitri 240	. 2 |- (x e. ~P{A, B, C} <-> x e. (({(/), {A}} u. {{B}, {A, B}}) u. ({{C}, {A, C}} u. {{B, C}, {A, B, C}})))
18	17	eqriv 2350	1 |- ~P{A, B, C} = (({(/), {A}} u. {{B}, {A, B}}) u. ({{C}, {A, C}} u. {{B, C}, {A, B, C}}))

Syntax hints:   \/ wo 357   = wceq 1642   e. wcel 1710   u. cun 3208   C_ wss 3258  (/)c0 3551  ~Pcpw 3723  {csn 3738  {cpr 3739  {ctp 3740

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by: (None)