Theorem alephsuc3 8854

Description: An alternate representation of a successor aleph. Compare alephsuc 6014 and alephsuc2 6029. Equality can be obtained by taking the card of the right-hand side then using alephcard 6015 and carden 5981.

Assertion

Ref	Expression
alephsuc3	|- (A e. On -> (aleph` suc A) ~~ {x e. On | x ~~ (aleph` A)})

Distinct variable group:   x,A

Proof of Theorem alephsuc3

Step	Hyp	Ref	Expression
1		alephsuc2 6029	. . . . . 6 |- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})
2	1	difeq1d 2725	. . . . 5 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ (aleph` A)))
3		alephcard 6015	. . . . . . . 8 |- (card` (aleph` A)) = (aleph` A)
4		cardval2 6007	. . . . . . . 8 |- (card` (aleph` A)) = {x e. On | x ~< (aleph` A)}
5	3, 4	eqtr3i 1910	. . . . . . 7 |- (aleph` A) = {x e. On | x ~< (aleph` A)}
6	5	a1i 8	. . . . . 6 |- (A e. On -> (aleph` A) = {x e. On | x ~< (aleph` A)})
7	6	difeq2d 2726	. . . . 5 |- (A e. On -> ({x e. On | x ~<_ (aleph` A)} \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}))
8	2, 7	eqtrd 1925	. . . 4 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}))
9		difrab 2868	. . . . 5 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
10		bren2 5448	. . . . . . 7 |- (x ~~ (aleph` A) <-> (x ~<_ (aleph` A) /\ -. x ~< (aleph` A)))
11	10	a1i 8	. . . . . 6 |- (x e. On -> (x ~~ (aleph` A) <-> (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))))
12	11	rabbiia 2285	. . . . 5 |- {x e. On | x ~~ (aleph` A)} = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
13	9, 12	eqtr4i 1911	. . . 4 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | x ~~ (aleph` A)}
14	8, 13	syl6req 1945	. . 3 |- (A e. On -> {x e. On | x ~~ (aleph` A)} = ((aleph` suc A) \ (aleph` A)))
15		sucelon 3898	. . . . . 6 |- (A e. On <-> suc A e. On)
16		alephgeom 6030	. . . . . 6 |- (suc A e. On <-> om C_ (aleph` suc A))
17	15, 16	bitri 190	. . . . 5 |- (A e. On <-> om C_ (aleph` suc A))
18		fvex 4689	. . . . . 6 |- (aleph` suc A) e. _V
19		ssdom2g 5468	. . . . . 6 |- ((aleph` suc A) e. _V -> (om C_ (aleph` suc A) -> om ~<_ (aleph` suc A)))
20	18, 19	ax-mp 7	. . . . 5 |- (om C_ (aleph` suc A) -> om ~<_ (aleph` suc A))
21	17, 20	sylbi 216	. . . 4 |- (A e. On -> om ~<_ (aleph` suc A))
22		alephordlem1 6020	. . . 4 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
23		fvex 4689	. . . . 5 |- (aleph` A) e. _V
24	18, 23	infdif 8837	. . . 4 |- ((om ~<_ (aleph` suc A) /\ (aleph` A) ~< (aleph` suc A)) -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
25	21, 22, 24	syl11anc 524	. . 3 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
26	14, 25	eqbrtrd 3357	. 2 |- (A e. On -> {x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A))
27	18	ensym 5471	. 2 |- ({x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A) -> (aleph` suc A) ~~ {x e. On | x ~~ (aleph` A)})
28	26, 27	syl 12	1 |- (A e. On -> (aleph` suc A) ~~ {x e. On | x ~~ (aleph` A)})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   \ cdif 2590   C_ wss 2593   class class class wbr 3338  Oncon0 3657  suc csuc 3659  omcom 3949  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859  alephcale 5860

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  ax-reg 5695  ax-inf2 5731  ax-ac 5906

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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  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-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  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-lim 3662  df-suc 3663  df-om 3950  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-iso 4015  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-undef 5556  df-riota 5560  df-card 5862  df-aleph 5863  df-cda 6066  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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