Theorem pw2en 5505

Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. (The proof seems excessively long. An attempt to find a shorter one is on my to-do list.)

Hypothesis

Ref	Expression
pw2en.1	|- A e. _V

Assertion

Ref	Expression
pw2en	|- ~PA ~~ (2o ^m A)

Proof of Theorem pw2en

Step	Hyp	Ref	Expression
1		pw2en.1	. . 3 |- A e. _V
2	1	pwex 3487	. 2 |- ~PA e. _V
3	1	opabex2 4539	. . 3 |- {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} e. _V
4	3	a1i 8	. 2 |- (x e. ~PA -> {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} e. _V)
5		visset 2295	. . . . 5 |- y e. _V
6	5	cnvex 4425	. . . 4 |- `'y e. _V
7		imaexg 4279	. . . 4 |- (`'y e. _V -> (`'y"{{(/)}}) e. _V)
8	6, 7	ax-mp 7	. . 3 |- (`'y"{{(/)}}) e. _V
9	8	a1i 8	. 2 |- (y e. (2o ^m A) -> (`'y"{{(/)}}) e. _V)
10		sseqin2 2811	. . . . . . . . 9 |- (x C_ A <-> (A i^i x) = x)
11	10	biimpi 168	. . . . . . . 8 |- (x C_ A -> (A i^i x) = x)
12	11	eqeq1d 1892	. . . . . . 7 |- (x C_ A -> ((A i^i x) = (`'y"{{(/)}}) <-> x = (`'y"{{(/)}})))
13		eleq2 1958	. . . . . . . . . 10 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (<.u, {(/)}>. e. y <-> <.u, {(/)}>. e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}))
14		p0ex 3495	. . . . . . . . . . . 12 |- {(/)} e. _V
15		visset 2295	. . . . . . . . . . . 12 |- u e. _V
16	14, 15	elimasn 4289	. . . . . . . . . . 11 |- (u e. (`'y"{{(/)}}) <-> <.{(/)}, u>. e. `'y)
17	14, 15	opelcnv 4143	. . . . . . . . . . 11 |- (<.{(/)}, u>. e. `'y <-> <.u, {(/)}>. e. y)
18	16, 17	bitri 190	. . . . . . . . . 10 |- (u e. (`'y"{{(/)}}) <-> <.u, {(/)}>. e. y)
19	13, 18	syl5bb 591	. . . . . . . . 9 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (u e. (`'y"{{(/)}}) <-> <.u, {(/)}>. e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}))
20		eq2ab 2004	. . . . . . . . . . . 12 |- ({v | v = (/)} = {v | (v = (/) /\ u e. x)} <-> A.v(v = (/) <-> (v = (/) /\ u e. x)))
21		df-sn 3049	. . . . . . . . . . . . 13 |- {(/)} = {v | v = (/)}
22	21	eqeq1i 1891	. . . . . . . . . . . 12 |- ({(/)} = {v | (v = (/) /\ u e. x)} <-> {v | v = (/)} = {v | (v = (/) /\ u e. x)})
23		iba 704	. . . . . . . . . . . . . 14 |- (u e. x -> (v = (/) <-> (v = (/) /\ u e. x)))
24	23	19.21aiv 1664	. . . . . . . . . . . . 13 |- (u e. x -> A.v(v = (/) <-> (v = (/) /\ u e. x)))
25		0ex 3446	. . . . . . . . . . . . . . 15 |- (/) e. _V
26		eqeq1 1890	. . . . . . . . . . . . . . . 16 |- (v = (/) -> (v = (/) <-> (/) = (/)))
27	26	anbi1d 679	. . . . . . . . . . . . . . . 16 |- (v = (/) -> ((v = (/) /\ u e. x) <-> ((/) = (/) /\ u e. x)))
28	26, 27	bibi12d 691	. . . . . . . . . . . . . . 15 |- (v = (/) -> ((v = (/) <-> (v = (/) /\ u e. x)) <-> ((/) = (/) <-> ((/) = (/) /\ u e. x))))
29	25, 28	cla4v 2370	. . . . . . . . . . . . . 14 |- (A.v(v = (/) <-> (v = (/) /\ u e. x)) -> ((/) = (/) <-> ((/) = (/) /\ u e. x)))
30		eqid 1884	. . . . . . . . . . . . . . . 16 |- (/) = (/)
31	30	a1bi 214	. . . . . . . . . . . . . . 15 |- (u e. x <-> ((/) = (/) -> u e. x))
32		pm4.71 697	. . . . . . . . . . . . . . 15 |- (((/) = (/) -> u e. x) <-> ((/) = (/) <-> ((/) = (/) /\ u e. x)))
33	31, 32	bitri 190	. . . . . . . . . . . . . 14 |- (u e. x <-> ((/) = (/) <-> ((/) = (/) /\ u e. x)))
34	29, 33	sylibr 217	. . . . . . . . . . . . 13 |- (A.v(v = (/) <-> (v = (/) /\ u e. x)) -> u e. x)
35	24, 34	impbii 174	. . . . . . . . . . . 12 |- (u e. x <-> A.v(v = (/) <-> (v = (/) /\ u e. x)))
36	20, 22, 35	3bitr4ri 201	. . . . . . . . . . 11 |- (u e. x <-> {(/)} = {v | (v = (/) /\ u e. x)})
37	36	anbi2i 538	. . . . . . . . . 10 |- ((u e. A /\ u e. x) <-> (u e. A /\ {(/)} = {v | (v = (/) /\ u e. x)}))
38		elin 2786	. . . . . . . . . 10 |- (u e. (A i^i x) <-> (u e. A /\ u e. x))
39		eleq1 1957	. . . . . . . . . . . 12 |- (z = u -> (z e. A <-> u e. A))
40		eleq1 1957	. . . . . . . . . . . . . . 15 |- (z = u -> (z e. x <-> u e. x))
41	40	anbi2d 678	. . . . . . . . . . . . . 14 |- (z = u -> ((v = (/) /\ z e. x) <-> (v = (/) /\ u e. x)))
42	41	abbidv 2008	. . . . . . . . . . . . 13 |- (z = u -> {v | (v = (/) /\ z e. x)} = {v | (v = (/) /\ u e. x)})
43	42	eqeq2d 1895	. . . . . . . . . . . 12 |- (z = u -> (w = {v | (v = (/) /\ z e. x)} <-> w = {v | (v = (/) /\ u e. x)}))
44	39, 43	anbi12d 690	. . . . . . . . . . 11 |- (z = u -> ((z e. A /\ w = {v | (v = (/) /\ z e. x)}) <-> (u e. A /\ w = {v | (v = (/) /\ u e. x)})))
45		eqeq1 1890	. . . . . . . . . . . 12 |- (w = {(/)} -> (w = {v | (v = (/) /\ u e. x)} <-> {(/)} = {v | (v = (/) /\ u e. x)}))
46	45	anbi2d 678	. . . . . . . . . . 11 |- (w = {(/)} -> ((u e. A /\ w = {v | (v = (/) /\ u e. x)}) <-> (u e. A /\ {(/)} = {v | (v = (/) /\ u e. x)})))
47	15, 14, 44, 46	opelopab 3570	. . . . . . . . . 10 |- (<.u, {(/)}>. e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> (u e. A /\ {(/)} = {v | (v = (/) /\ u e. x)}))
48	37, 38, 47	3bitr4i 200	. . . . . . . . 9 |- (u e. (A i^i x) <-> <.u, {(/)}>. e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})})
49	19, 48	syl6rbbr 598	. . . . . . . 8 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (u e. (A i^i x) <-> u e. (`'y"{{(/)}})))
50	49	eqrdv 1882	. . . . . . 7 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (A i^i x) = (`'y"{{(/)}}))
51	12, 50	syl5cbi 226	. . . . . 6 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (x C_ A -> x = (`'y"{{(/)}})))
52		sseq1 2637	. . . . . . 7 |- (x = (`'y"{{(/)}}) -> (x C_ A <-> (`'y"{{(/)}}) C_ A))
53		imassrn 4278	. . . . . . . 8 |- (`'y"{{(/)}}) C_ ran `' y
54	21, 14	eqeltrri 1968	. . . . . . . . . . . . . 14 |- {v | v = (/)} e. _V
55		simpl 346	. . . . . . . . . . . . . . 15 |- ((v = (/) /\ z e. x) -> v = (/))
56	55	ss2abi 2679	. . . . . . . . . . . . . 14 |- {v | (v = (/) /\ z e. x)} C_ {v | v = (/)}
57	54, 56	ssexi 3456	. . . . . . . . . . . . 13 |- {v | (v = (/) /\ z e. x)} e. _V
58		eqid 1884	. . . . . . . . . . . . 13 |- {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}
59	57, 58	fnopab2 4549	. . . . . . . . . . . 12 |- {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} Fn A
60		fneq1 4503	. . . . . . . . . . . 12 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (y Fn A <-> {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} Fn A))
61	59, 60	mpbiri 211	. . . . . . . . . . 11 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> y Fn A)
62		fndm 4512	. . . . . . . . . . 11 |- (y Fn A -> dom y = A)
63	61, 62	syl 12	. . . . . . . . . 10 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> dom y = A)
64		dfdm4 4151	. . . . . . . . . 10 |- dom y = ran `' y
65	63, 64	syl5eqr 1942	. . . . . . . . 9 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> ran `' y = A)
66	65	sseq2d 2645	. . . . . . . 8 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> ((`'y"{{(/)}}) C_ ran `' y <-> (`'y"{{(/)}}) C_ A))
67	53, 66	mpbii 210	. . . . . . 7 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (`'y"{{(/)}}) C_ A)
68	52, 67	syl5cbir 228	. . . . . 6 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (x = (`'y"{{(/)}}) -> x C_ A))
69	51, 68	impbid 574	. . . . 5 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (x C_ A <-> x = (`'y"{{(/)}})))
70		visset 2295	. . . . . 6 |- x e. _V
71	70	elpw 3037	. . . . 5 |- (x e. ~PA <-> x C_ A)
72	69, 71	syl5bb 591	. . . 4 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (x e. ~PA <-> x = (`'y"{{(/)}})))
73	72	pm5.32ri 708	. . 3 |- ((x e. ~PA /\ y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}) <-> (x = (`'y"{{(/)}}) /\ y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}))
74		ancom 482	. . 3 |- ((x = (`'y"{{(/)}}) /\ y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}) <-> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} /\ x = (`'y"{{(/)}})))
75		iba 704	. . . . . . . . . . . . 13 |- (z e. x -> (v = (/) <-> (v = (/) /\ z e. x)))
76		elsn 3058	. . . . . . . . . . . . 13 |- (v e. {(/)} <-> v = (/))
77	75, 76	syl5bb 591	. . . . . . . . . . . 12 |- (z e. x -> (v e. {(/)} <-> (v = (/) /\ z e. x)))
78	77	abbi2dv 2009	. . . . . . . . . . 11 |- (z e. x -> {(/)} = {v | (v = (/) /\ z e. x)})
79	14	prid2 3107	. . . . . . . . . . . 12 |- {(/)} e. {(/), {(/)}}
80		df2o2 5186	. . . . . . . . . . . 12 |- 2o = {(/), {(/)}}
81	79, 80	eleqtrri 1970	. . . . . . . . . . 11 |- {(/)} e. 2o
82	78, 81	syl6eqelr 1980	. . . . . . . . . 10 |- (z e. x -> {v | (v = (/) /\ z e. x)} e. 2o)
83		abn0 2892	. . . . . . . . . . . . 13 |- ({v | (v = (/) /\ z e. x)} =/= (/) <-> E.v(v = (/) /\ z e. x))
84		simpr 350	. . . . . . . . . . . . . 14 |- ((v = (/) /\ z e. x) -> z e. x)
85	84	19.23aiv 1674	. . . . . . . . . . . . 13 |- (E.v(v = (/) /\ z e. x) -> z e. x)
86	83, 85	sylbi 216	. . . . . . . . . . . 12 |- ({v | (v = (/) /\ z e. x)} =/= (/) -> z e. x)
87	86	necon1bi 2048	. . . . . . . . . . 11 |- (-. z e. x -> {v | (v = (/) /\ z e. x)} = (/))
88	25	prid1 3106	. . . . . . . . . . . 12 |- (/) e. {(/), {(/)}}
89	88, 80	eleqtrri 1970	. . . . . . . . . . 11 |- (/) e. 2o
90	87, 89	syl6eqel 1979	. . . . . . . . . 10 |- (-. z e. x -> {v | (v = (/) /\ z e. x)} e. 2o)
91	82, 90	pm2.61i 140	. . . . . . . . 9 |- {v | (v = (/) /\ z e. x)} e. 2o
92	91	a1i 8	. . . . . . . 8 |- (z e. A -> {v | (v = (/) /\ z e. x)} e. 2o)
93	58, 92	fopab 4800	. . . . . . 7 |- {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}:A-->2o
94		feq1 4551	. . . . . . 7 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> (y:A-->2o <-> {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}:A-->2o))
95	93, 94	mpbiri 211	. . . . . 6 |- (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> y:A-->2o)
96		eleq2 1958	. . . . . . . . . . . . . . . 16 |- (x = (`'y"{{(/)}}) -> (z e. x <-> z e. (`'y"{{(/)}})))
97		visset 2295	. . . . . . . . . . . . . . . . . 18 |- z e. _V
98	14, 97	elimasn 4289	. . . . . . . . . . . . . . . . 17 |- (z e. (`'y"{{(/)}}) <-> <.{(/)}, z>. e. `'y)
99	14, 97	opelcnv 4143	. . . . . . . . . . . . . . . . 17 |- (<.{(/)}, z>. e. `'y <-> <.z, {(/)}>. e. y)
100	98, 99	bitri 190	. . . . . . . . . . . . . . . 16 |- (z e. (`'y"{{(/)}}) <-> <.z, {(/)}>. e. y)
101	96, 100	syl6bb 595	. . . . . . . . . . . . . . 15 |- (x = (`'y"{{(/)}}) -> (z e. x <-> <.z, {(/)}>. e. y))
102	14	fnopfvb 4713	. . . . . . . . . . . . . . . . 17 |- ((y Fn A /\ z e. A) -> ((y` z) = {(/)} <-> <.z, {(/)}>. e. y))
103		ffn 4562	. . . . . . . . . . . . . . . . 17 |- (y:A-->2o -> y Fn A)
104	102, 103	sylan 497	. . . . . . . . . . . . . . . 16 |- ((y:A-->2o /\ z e. A) -> ((y` z) = {(/)} <-> <.z, {(/)}>. e. y))
105	104	bicomd 580	. . . . . . . . . . . . . . 15 |- ((y:A-->2o /\ z e. A) -> (<.z, {(/)}>. e. y <-> (y` z) = {(/)}))
106	101, 105	sylan9bb 599	. . . . . . . . . . . . . 14 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (z e. x <-> (y` z) = {(/)}))
107	106	biimpa 460	. . . . . . . . . . . . 13 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ z e. x) -> (y` z) = {(/)})
108	78	adantl 424	. . . . . . . . . . . . 13 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ z e. x) -> {(/)} = {v | (v = (/) /\ z e. x)})
109	107, 108	eqtrd 1925	. . . . . . . . . . . 12 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ z e. x) -> (y` z) = {v | (v = (/) /\ z e. x)})
110		ffvelrn 4787	. . . . . . . . . . . . . . . . . . 19 |- ((y:A-->2o /\ z e. A) -> (y` z) e. 2o)
111	80	eleq2i 1961	. . . . . . . . . . . . . . . . . . . 20 |- ((y` z) e. 2o <-> (y` z) e. {(/), {(/)}})
112		fvex 4689	. . . . . . . . . . . . . . . . . . . . 21 |- (y` z) e. _V
113	112	elpr 3061	. . . . . . . . . . . . . . . . . . . 20 |- ((y` z) e. {(/), {(/)}} <-> ((y` z) = (/) \/ (y` z) = {(/)}))
114	111, 113	bitri 190	. . . . . . . . . . . . . . . . . . 19 |- ((y` z) e. 2o <-> ((y` z) = (/) \/ (y` z) = {(/)}))
115	110, 114	sylib 215	. . . . . . . . . . . . . . . . . 18 |- ((y:A-->2o /\ z e. A) -> ((y` z) = (/) \/ (y` z) = {(/)}))
116	115	ord 249	. . . . . . . . . . . . . . . . 17 |- ((y:A-->2o /\ z e. A) -> (-. (y` z) = (/) -> (y` z) = {(/)}))
117	116	adantl 424	. . . . . . . . . . . . . . . 16 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (-. (y` z) = (/) -> (y` z) = {(/)}))
118	117, 106	sylibrd 221	. . . . . . . . . . . . . . 15 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (-. (y` z) = (/) -> z e. x))
119	118	con1d 109	. . . . . . . . . . . . . 14 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (-. z e. x -> (y` z) = (/)))
120	119	imp 377	. . . . . . . . . . . . 13 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ -. z e. x) -> (y` z) = (/))
121	87	adantl 424	. . . . . . . . . . . . 13 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ -. z e. x) -> {v | (v = (/) /\ z e. x)} = (/))
122	120, 121	eqtr4d 1928	. . . . . . . . . . . 12 |- (((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) /\ -. z e. x) -> (y` z) = {v | (v = (/) /\ z e. x)})
123	109, 122	pm2.61dan 535	. . . . . . . . . . 11 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (y` z) = {v | (v = (/) /\ z e. x)})
124		fvopab2 4754	. . . . . . . . . . . . 13 |- ((z e. A /\ {v | (v = (/) /\ z e. x)} e. _V) -> ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z) = {v | (v = (/) /\ z e. x)})
125	57, 124	mpan2 760	. . . . . . . . . . . 12 |- (z e. A -> ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z) = {v | (v = (/) /\ z e. x)})
126	125	ad2antll 443	. . . . . . . . . . 11 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z) = {v | (v = (/) /\ z e. x)})
127	123, 126	eqtr4d 1928	. . . . . . . . . 10 |- ((x = (`'y"{{(/)}}) /\ (y:A-->2o /\ z e. A)) -> (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z))
128	127	expr 418	. . . . . . . . 9 |- ((x = (`'y"{{(/)}}) /\ y:A-->2o) -> (z e. A -> (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z)))
129	128	r19.21aiv 2175	. . . . . . . 8 |- ((x = (`'y"{{(/)}}) /\ y:A-->2o) -> A.z e. A (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z))
130		ax-17 1317	. . . . . . . . . . 11 |- (u e. y -> A.z u e. y)
131		hbopab1 3562	. . . . . . . . . . 11 |- (u e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} -> A.z u e. {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})})
132	130, 131	eqfnfv2f 4770	. . . . . . . . . 10 |- ((y Fn A /\ {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} Fn A) -> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> A.z e. A (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z)))
133	132, 103, 59	sylancl 525	. . . . . . . . 9 |- (y:A-->2o -> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> A.z e. A (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z)))
134	133	adantl 424	. . . . . . . 8 |- ((x = (`'y"{{(/)}}) /\ y:A-->2o) -> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> A.z e. A (y` z) = ({<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}` z)))
135	129, 134	mpbird 213	. . . . . . 7 |- ((x = (`'y"{{(/)}}) /\ y:A-->2o) -> y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})})
136	135	ex 402	. . . . . 6 |- (x = (`'y"{{(/)}}) -> (y:A-->2o -> y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}))
137	95, 136	impbid2 576	. . . . 5 |- (x = (`'y"{{(/)}}) -> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> y:A-->2o))
138		2on 5183	. . . . . . 7 |- 2o e. On
139	138	elisseti 2301	. . . . . 6 |- 2o e. _V
140	139, 1	elmap 5393	. . . . 5 |- (y e. (2o ^m A) <-> y:A-->2o)
141	137, 140	syl6bbr 597	. . . 4 |- (x = (`'y"{{(/)}}) -> (y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} <-> y e. (2o ^m A)))
142	141	pm5.32ri 708	. . 3 |- ((y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})} /\ x = (`'y"{{(/)}})) <-> (y e. (2o ^m A) /\ x = (`'y"{{(/)}})))
143	73, 74, 142	3bitri 194	. 2 |- ((x e. ~PA /\ y = {<.z, w>. | (z e. A /\ w = {v | (v = (/) /\ z e. x)})}) <-> (y e. (2o ^m A) /\ x = (`'y"{{(/)}})))
144	2, 4, 9, 143	en2 5461	1 |- ~PA ~~ (2o ^m A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  {cpr 3045  <.cop 3046   class class class wbr 3338  {copab 3395  Oncon0 3657  `'ccnv 3985  dom cdm 3986  ran crn 3987  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  2oc2o 5173   ^m cmap 5381   ~~ cen 5423

This theorem is referenced by:  pwen 5597  aleph1 6019  infmap1 8842  pw2eng 14419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-2o 5178  df-map 5383  df-en 5427

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