Theorem ce2 6193

Description: The value of base two cardinal exponentiation. Theorem XI.2.70 of [Rosser] p. 389. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
ce2.1	|- A e. _V

Assertion

Ref	Expression
ce2	|- (M = Nc ~P1 A -> (2c ↑c M) = Nc ~PA)

Proof of Theorem ce2

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

Step	Hyp	Ref	Expression
1		oveq2 5532	. 2 |- (M = Nc ~P1 A -> (2c ↑c M) = (2c ↑c Nc ~P1 A))
2		df-pr 3743	. . . . . . . . . 10 |- {_V, (/)} = ({_V} u. {(/)})
3		pw1eq 4144	. . . . . . . . . 10 |- ({_V, (/)} = ({_V} u. {(/)}) -> ~P1 {_V, (/)} = ~P1 ({_V} u. {(/)}))
4	2, 3	ax-mp 5	. . . . . . . . 9 |- ~P1 {_V, (/)} = ~P1 ({_V} u. {(/)})
5		pw1un 4164	. . . . . . . . 9 |- ~P1 ({_V} u. {(/)}) = (~P1 {_V} u. ~P1 {(/)})
6	4, 5	eqtri 2373	. . . . . . . 8 |- ~P1 {_V, (/)} = (~P1 {_V} u. ~P1 {(/)})
7		df-pr 3743	. . . . . . . . 9 |- {{_V}, {(/)}} = ({{_V}} u. {{(/)}})
8		vvex 4110	. . . . . . . . . . 11 |- _V e. _V
9	8	pw1sn 4166	. . . . . . . . . 10 |- ~P1 {_V} = {{_V}}
10		0ex 4111	. . . . . . . . . . 11 |- (/) e. _V
11	10	pw1sn 4166	. . . . . . . . . 10 |- ~P1 {(/)} = {{(/)}}
12	9, 11	uneq12i 3417	. . . . . . . . 9 |- (~P1 {_V} u. ~P1 {(/)}) = ({{_V}} u. {{(/)}})
13	7, 12	eqtr4i 2376	. . . . . . . 8 |- {{_V}, {(/)}} = (~P1 {_V} u. ~P1 {(/)})
14	6, 13	eqtr4i 2376	. . . . . . 7 |- ~P1 {_V, (/)} = {{_V}, {(/)}}
15		vn0 3558	. . . . . . . . . 10 |- _V =/= (/)
16	8	sneqb 3877	. . . . . . . . . . 11 |- ({_V} = {(/)} <-> _V = (/))
17	16	necon3bii 2549	. . . . . . . . . 10 |- ({_V} =/= {(/)} <-> _V =/= (/))
18	15, 17	mpbir 200	. . . . . . . . 9 |- {_V} =/= {(/)}
19		eqid 2353	. . . . . . . . 9 |- {{_V}, {(/)}} = {{_V}, {(/)}}
20		snex 4112	. . . . . . . . . 10 |- {_V} e. _V
21		snex 4112	. . . . . . . . . 10 |- {(/)} e. _V
22		neeq1 2525	. . . . . . . . . . . 12 |- (x = {_V} -> (x =/= y <-> {_V} =/= y))
23		neeq2 2526	. . . . . . . . . . . 12 |- (y = {(/)} -> ({_V} =/= y <-> {_V} =/= {(/)}))
24	22, 23	sylan9bb 680	. . . . . . . . . . 11 |- ((x = {_V} /\ y = {(/)}) -> (x =/= y <-> {_V} =/= {(/)}))
25		preq12 3802	. . . . . . . . . . . 12 |- ((x = {_V} /\ y = {(/)}) -> {x, y} = {{_V}, {(/)}})
26	25	eqeq2d 2364	. . . . . . . . . . 11 |- ((x = {_V} /\ y = {(/)}) -> ({{_V}, {(/)}} = {x, y} <-> {{_V}, {(/)}} = {{_V}, {(/)}}))
27	24, 26	anbi12d 691	. . . . . . . . . 10 |- ((x = {_V} /\ y = {(/)}) -> ((x =/= y /\ {{_V}, {(/)}} = {x, y}) <-> ({_V} =/= {(/)} /\ {{_V}, {(/)}} = {{_V}, {(/)}})))
28	20, 21, 27	spc2ev 2948	. . . . . . . . 9 |- (({_V} =/= {(/)} /\ {{_V}, {(/)}} = {{_V}, {(/)}}) -> E.xE.y(x =/= y /\ {{_V}, {(/)}} = {x, y}))
29	18, 19, 28	mp2an 653	. . . . . . . 8 |- E.xE.y(x =/= y /\ {{_V}, {(/)}} = {x, y})
30		el2c 6192	. . . . . . . 8 |- ({{_V}, {(/)}} e. 2c <-> E.xE.y(x =/= y /\ {{_V}, {(/)}} = {x, y}))
31	29, 30	mpbir 200	. . . . . . 7 |- {{_V}, {(/)}} e. 2c
32	14, 31	eqeltri 2423	. . . . . 6 |- ~P1 {_V, (/)} e. 2c
33		2nc 6169	. . . . . . 7 |- 2c e. NC
34		ncseqnc 6129	. . . . . . 7 |- (2c e. NC -> (2c = Nc ~P1 {_V, (/)} <-> ~P1 {_V, (/)} e. 2c))
35	33, 34	ax-mp 5	. . . . . 6 |- (2c = Nc ~P1 {_V, (/)} <-> ~P1 {_V, (/)} e. 2c)
36	32, 35	mpbir 200	. . . . 5 |- 2c = Nc ~P1 {_V, (/)}
37	36	oveq1i 5534	. . . 4 |- (2c ↑c Nc ~P1 A) = ( Nc ~P1 {_V, (/)} ↑c Nc ~P1 A)
38		prex 4113	. . . . 5 |- {_V, (/)} e. _V
39		ce2.1	. . . . 5 |- A e. _V
40	38, 39	cenc 6182	. . . 4 |- ( Nc ~P1 {_V, (/)} ↑c Nc ~P1 A) = Nc ({_V, (/)} ^m A)
41	37, 40	eqtri 2373	. . 3 |- (2c ↑c Nc ~P1 A) = Nc ({_V, (/)} ^m A)
42		eqid 2353	. . . . 5 |- {_V, (/)} = {_V, (/)}
43	8, 10, 39	enprmapc 6084	. . . . 5 |- ((_V =/= (/) /\ {_V, (/)} = {_V, (/)}) -> ({_V, (/)} ^m A) ~~ ~PA)
44	15, 42, 43	mp2an 653	. . . 4 |- ({_V, (/)} ^m A) ~~ ~PA
45		ovex 5552	. . . . 5 |- ({_V, (/)} ^m A) e. _V
46	45	eqnc 6128	. . . 4 |- ( Nc ({_V, (/)} ^m A) = Nc ~PA <-> ({_V, (/)} ^m A) ~~ ~PA)
47	44, 46	mpbir 200	. . 3 |- Nc ({_V, (/)} ^m A) = Nc ~PA
48	41, 47	eqtri 2373	. 2 |- (2c ↑c Nc ~P1 A) = Nc ~PA
49	1, 48	syl6eq 2401	1 |- (M = Nc ~P1 A -> (2c ↑c M) = Nc ~PA)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   =/= wne 2517  _Vcvv 2860   u. cun 3208  (/)c0 3551  ~Pcpw 3723  {csn 3738  {cpr 3739  ~P₁ cpw1 4136   class class class wbr 4640  (class class class)co 5526   ^m cmap 6000   ~~ cen 6029   NC cncs 6089   Nc cnc 6092  2_cc2c 6095   ↑_c cce 6097

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107

This theorem is referenced by:  ce2nc1  6194  ce2ncpw11c  6195  ce2lt  6221  ce2le  6234  tce2  6237