Theorem alephexp2 7528

Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 7526 (which works if the base is less than or equal to the exponent) and infmap2 7523 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result.

Assertion

Ref	Expression
alephexp2	|- (A e. On -> (2o ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))})

Distinct variable group:   x,A

Proof of Theorem alephexp2

Step	Hyp	Ref	Expression
1		alephgeom 4854	. . . . 5 |- (A e. On <-> om (_ (aleph` A))
2		fvex 3717	. . . . . 6 |- (aleph` A) e. V
3		ssdom2g 4390	. . . . . 6 |- ((aleph` A) e. V -> (om (_ (aleph` A) -> om ~<_ (aleph` A)))
4	2, 3	ax-mp 7	. . . . 5 |- (om (_ (aleph` A) -> om ~<_ (aleph` A))
5	1, 4	sylbi 199	. . . 4 |- (A e. On -> om ~<_ (aleph` A))
6		domrefg 4374	. . . . 5 |- ((aleph` A) e. V -> (aleph` A) ~<_ (aleph` A))
7	2, 6	ax-mp 7	. . . 4 |- (aleph` A) ~<_ (aleph` A)
8	5, 7	jctir 293	. . 3 |- (A e. On -> (om ~<_ (aleph` A) /\ (aleph` A) ~<_ (aleph` A)))
9	2, 2	infmap2 7523	. . 3 |- ((om ~<_ (aleph` A) /\ (aleph` A) ~<_ (aleph` A)) -> ((aleph` A) ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))})
10	8, 9	syl 10	. 2 |- (A e. On -> ((aleph` A) ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))})
11		pm3.2 283	. . . . 5 |- (A e. On -> (A e. On -> (A e. On /\ A e. On)))
12	11	pm2.43i 64	. . . 4 |- (A e. On -> (A e. On /\ A e. On))
13		ssid 2070	. . . 4 |- A (_ A
14	12, 13	jctir 293	. . 3 |- (A e. On -> ((A e. On /\ A e. On) /\ A (_ A))
15		alephexp1 7526	. . 3 |- (((A e. On /\ A e. On) /\ A (_ A) -> ((aleph` A) ^m (aleph` A)) ~~ (2o ^m (aleph` A)))
16		oprex 3968	. . . 4 |- (2o ^m (aleph` A)) e. V
17		enen1 4457	. . . 4 |- (((2o ^m (aleph` A)) e. V /\ ((aleph` A) ^m (aleph` A)) ~~ (2o ^m (aleph` A))) -> (((aleph` A) ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))} <-> (2o ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))}))
18	16, 17	mpan 693	. . 3 |- (((aleph` A) ^m (aleph` A)) ~~ (2o ^m (aleph` A)) -> (((aleph` A) ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))} <-> (2o ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))}))
19	14, 15, 18	3syl 20	. 2 |- (A e. On -> (((aleph` A) ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))} <-> (2o ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))}))
20	10, 19	mpbid 195	1 |- (A e. On -> (2o ^m (aleph` A)) ~~ {x | (x (_ (aleph` A) /\ x ~~ (aleph` A))})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955  {cab 1456  Vcvv 1802   (_ wss 2037   class class class wbr 2609  Oncon0 2938  omcom 3121  ` cfv 3172  (class class class)co 3948  2oc2o 4113   ^m cm 4306   ~~ cen 4348   ~<_ cdom 4349  alephcale 4786

This theorem is referenced by:  gch-kn 7529

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-iso 3189  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-2o 4118  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-seq1 6245  df-exp 6501

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