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 ∈ V

Assertion

Ref	Expression
ce2	⊢ (M = Nc ℘1A → (2c ↑c M) = Nc ℘A)

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 ℘1A → (2c ↑c M) = (2c ↑c Nc ℘1A))
2		df-pr 3743	. . . . . . . . . 10 ⊢ {V, ∅} = ({V} ∪ {∅})
3		pw1eq 4144	. . . . . . . . . 10 ⊢ ({V, ∅} = ({V} ∪ {∅}) → ℘1{V, ∅} = ℘1({V} ∪ {∅}))
4	2, 3	ax-mp 5	. . . . . . . . 9 ⊢ ℘1{V, ∅} = ℘1({V} ∪ {∅})
5		pw1un 4164	. . . . . . . . 9 ⊢ ℘1({V} ∪ {∅}) = (℘1{V} ∪ ℘1{∅})
6	4, 5	eqtri 2373	. . . . . . . 8 ⊢ ℘1{V, ∅} = (℘1{V} ∪ ℘1{∅})
7		df-pr 3743	. . . . . . . . 9 ⊢ {{V}, {∅}} = ({{V}} ∪ {{∅}})
8		vvex 4110	. . . . . . . . . . 11 ⊢ V ∈ V
9	8	pw1sn 4166	. . . . . . . . . 10 ⊢ ℘1{V} = {{V}}
10		0ex 4111	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	pw1sn 4166	. . . . . . . . . 10 ⊢ ℘1{∅} = {{∅}}
12	9, 11	uneq12i 3417	. . . . . . . . 9 ⊢ (℘1{V} ∪ ℘1{∅}) = ({{V}} ∪ {{∅}})
13	7, 12	eqtr4i 2376	. . . . . . . 8 ⊢ {{V}, {∅}} = (℘1{V} ∪ ℘1{∅})
14	6, 13	eqtr4i 2376	. . . . . . 7 ⊢ ℘1{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} ∈ V
21		snex 4112	. . . . . . . . . 10 ⊢ {∅} ∈ 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}, {∅}}) → ∃x∃y(x ≠ y ∧ {{V}, {∅}} = {x, y}))
29	18, 19, 28	mp2an 653	. . . . . . . 8 ⊢ ∃x∃y(x ≠ y ∧ {{V}, {∅}} = {x, y})
30		el2c 6192	. . . . . . . 8 ⊢ ({{V}, {∅}} ∈ 2c ↔ ∃x∃y(x ≠ y ∧ {{V}, {∅}} = {x, y}))
31	29, 30	mpbir 200	. . . . . . 7 ⊢ {{V}, {∅}} ∈ 2c
32	14, 31	eqeltri 2423	. . . . . 6 ⊢ ℘1{V, ∅} ∈ 2c
33		2nc 6169	. . . . . . 7 ⊢ 2c ∈ NC
34		ncseqnc 6129	. . . . . . 7 ⊢ (2c ∈ NC → (2c = Nc ℘1{V, ∅} ↔ ℘1{V, ∅} ∈ 2c))
35	33, 34	ax-mp 5	. . . . . 6 ⊢ (2c = Nc ℘1{V, ∅} ↔ ℘1{V, ∅} ∈ 2c)
36	32, 35	mpbir 200	. . . . 5 ⊢ 2c = Nc ℘1{V, ∅}
37	36	oveq1i 5534	. . . 4 ⊢ (2c ↑c Nc ℘1A) = ( Nc ℘1{V, ∅} ↑c Nc ℘1A)
38		prex 4113	. . . . 5 ⊢ {V, ∅} ∈ V
39		ce2.1	. . . . 5 ⊢ A ∈ V
40	38, 39	cenc 6182	. . . 4 ⊢ ( Nc ℘1{V, ∅} ↑c Nc ℘1A) = Nc ({V, ∅} ↑m A)
41	37, 40	eqtri 2373	. . 3 ⊢ (2c ↑c Nc ℘1A) = Nc ({V, ∅} ↑m A)
42		eqid 2353	. . . . 5 ⊢ {V, ∅} = {V, ∅}
43	8, 10, 39	enprmapc 6084	. . . . 5 ⊢ ((V ≠ ∅ ∧ {V, ∅} = {V, ∅}) → ({V, ∅} ↑m A) ≈ ℘A)
44	15, 42, 43	mp2an 653	. . . 4 ⊢ ({V, ∅} ↑m A) ≈ ℘A
45		ovex 5552	. . . . 5 ⊢ ({V, ∅} ↑m A) ∈ V
46	45	eqnc 6128	. . . 4 ⊢ ( Nc ({V, ∅} ↑m A) = Nc ℘A ↔ ({V, ∅} ↑m A) ≈ ℘A)
47	44, 46	mpbir 200	. . 3 ⊢ Nc ({V, ∅} ↑m A) = Nc ℘A
48	41, 47	eqtri 2373	. 2 ⊢ (2c ↑c Nc ℘1A) = Nc ℘A
49	1, 48	syl6eq 2401	1 ⊢ (M = Nc ℘1A → (2c ↑c M) = Nc ℘A)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860   ∪ cun 3208  ∅c0 3551  ℘cpw 3723  {csn 3738  {cpr 3739  ℘₁cpw1 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