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Theorem ce2 6192
 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 → (2cc M) = Nc A)

Proof of Theorem ce2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5531 . 2 (M = Nc 1A → (2cc M) = (2cc Nc 1A))
2 df-pr 3742 . . . . . . . . . 10 {V, } = ({V} ∪ {})
3 pw1eq 4143 . . . . . . . . . 10 ({V, } = ({V} ∪ {}) → 1{V, } = 1({V} ∪ {}))
42, 3ax-mp 5 . . . . . . . . 9 1{V, } = 1({V} ∪ {})
5 pw1un 4163 . . . . . . . . 9 1({V} ∪ {}) = (1{V} ∪ 1{})
64, 5eqtri 2373 . . . . . . . 8 1{V, } = (1{V} ∪ 1{})
7 df-pr 3742 . . . . . . . . 9 {{V}, {}} = ({{V}} ∪ {{}})
8 vvex 4109 . . . . . . . . . . 11 V V
98pw1sn 4165 . . . . . . . . . 10 1{V} = {{V}}
10 0ex 4110 . . . . . . . . . . 11 V
1110pw1sn 4165 . . . . . . . . . 10 1{} = {{}}
129, 11uneq12i 3416 . . . . . . . . 9 (1{V} ∪ 1{}) = ({{V}} ∪ {{}})
137, 12eqtr4i 2376 . . . . . . . 8 {{V}, {}} = (1{V} ∪ 1{})
146, 13eqtr4i 2376 . . . . . . 7 1{V, } = {{V}, {}}
15 vn0 3557 . . . . . . . . . 10 V ≠
168sneqb 3876 . . . . . . . . . . 11 ({V} = {} ↔ V = )
1716necon3bii 2548 . . . . . . . . . 10 ({V} ≠ {} ↔ V ≠ )
1815, 17mpbir 200 . . . . . . . . 9 {V} ≠ {}
19 eqid 2353 . . . . . . . . 9 {{V}, {}} = {{V}, {}}
20 snex 4111 . . . . . . . . . 10 {V} V
21 snex 4111 . . . . . . . . . 10 {} V
22 neeq1 2524 . . . . . . . . . . . 12 (x = {V} → (xy ↔ {V} ≠ y))
23 neeq2 2525 . . . . . . . . . . . 12 (y = {} → ({V} ≠ y ↔ {V} ≠ {}))
2422, 23sylan9bb 680 . . . . . . . . . . 11 ((x = {V} y = {}) → (xy ↔ {V} ≠ {}))
25 preq12 3801 . . . . . . . . . . . 12 ((x = {V} y = {}) → {x, y} = {{V}, {}})
2625eqeq2d 2364 . . . . . . . . . . 11 ((x = {V} y = {}) → ({{V}, {}} = {x, y} ↔ {{V}, {}} = {{V}, {}}))
2724, 26anbi12d 691 . . . . . . . . . 10 ((x = {V} y = {}) → ((xy {{V}, {}} = {x, y}) ↔ ({V} ≠ {} {{V}, {}} = {{V}, {}})))
2820, 21, 27spc2ev 2947 . . . . . . . . 9 (({V} ≠ {} {{V}, {}} = {{V}, {}}) → xy(xy {{V}, {}} = {x, y}))
2918, 19, 28mp2an 653 . . . . . . . 8 xy(xy {{V}, {}} = {x, y})
30 el2c 6191 . . . . . . . 8 ({{V}, {}} 2cxy(xy {{V}, {}} = {x, y}))
3129, 30mpbir 200 . . . . . . 7 {{V}, {}} 2c
3214, 31eqeltri 2423 . . . . . 6 1{V, } 2c
33 2nc 6168 . . . . . . 7 2c NC
34 ncseqnc 6128 . . . . . . 7 (2c NC → (2c = Nc 1{V, } ↔ 1{V, } 2c))
3533, 34ax-mp 5 . . . . . 6 (2c = Nc 1{V, } ↔ 1{V, } 2c)
3632, 35mpbir 200 . . . . 5 2c = Nc 1{V, }
3736oveq1i 5533 . . . 4 (2cc Nc 1A) = ( Nc 1{V, } ↑c Nc 1A)
38 prex 4112 . . . . 5 {V, } V
39 ce2.1 . . . . 5 A V
4038, 39cenc 6181 . . . 4 ( Nc 1{V, } ↑c Nc 1A) = Nc ({V, } ↑m A)
4137, 40eqtri 2373 . . 3 (2cc Nc 1A) = Nc ({V, } ↑m A)
42 eqid 2353 . . . . 5 {V, } = {V, }
438, 10, 39enprmapc 6083 . . . . 5 ((V ≠ {V, } = {V, }) → ({V, } ↑m A) ≈ A)
4415, 42, 43mp2an 653 . . . 4 ({V, } ↑m A) ≈ A
45 ovex 5551 . . . . 5 ({V, } ↑m A) V
4645eqnc 6127 . . . 4 ( Nc ({V, } ↑m A) = Nc A ↔ ({V, } ↑m A) ≈ A)
4744, 46mpbir 200 . . 3 Nc ({V, } ↑m A) = Nc A
4841, 47eqtri 2373 . 2 (2cc Nc 1A) = Nc A
491, 48syl6eq 2401 1 (M = Nc 1A → (2cc M) = Nc A)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ∪ cun 3207  ∅c0 3550  ℘cpw 3722  {csn 3737  {cpr 3738  ℘1cpw1 4135   class class class wbr 4639  (class class class)co 5525   ↑m cmap 5999   ≈ cen 6028   NC cncs 6088   Nc cnc 6091  2cc2c 6094   ↑c cce 6096 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-2c 6104  df-ce 6106 This theorem is referenced by:  ce2nc1  6193  ce2ncpw11c  6194  ce2lt  6220  ce2le  6233  tce2  6236
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