Theorem alephsuc3 4955

Description: An alternate representation of a successor aleph. Compare alephsuc 3672 and alephsuc2 3686. Equality can be obtained by taking the card of the right-hand side then using alephcard 3673 and carden 3638.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem alephsuc3

Step	Hyp	Ref	Expression
1		alephsuc2 3686	. . . . . 6 |- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})
2	1	difeq1d 1587	. . . . 5 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ (aleph` A)))
3		alephcard 3673	. . . . . . . 8 |- (card` (aleph` A)) = (aleph` A)
4		cardval2 3661	. . . . . . . 8 |- (card` (aleph` A)) = {x e. On | x ~< (aleph` A)}
5	3, 4	eqtr3 1121	. . . . . . 7 |- (aleph` A) = {x e. On | x ~< (aleph` A)}
6	5	a1i 7	. . . . . 6 |- (A e. On -> (aleph` A) = {x e. On | x ~< (aleph` A)})
7	6	difeq2d 1588	. . . . 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 1128	. . . 4 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}))
9		difrab 1695	. . . . 5 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
10		bren2 3293	. . . . . . 7 |- (x ~~ (aleph` A) <-> (x ~<_ (aleph` A) /\ -. x ~< (aleph` A)))
11	10	a1i 7	. . . . . 6 |- (x e. On -> (x ~~ (aleph` A) <-> (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))))
12	11	birabi 1342	. . . . 5 |- {x e. On | x ~~ (aleph` A)} = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
13	9, 12	eqtr4 1122	. . . 4 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | x ~~ (aleph` A)}
14	8, 13	syl6req 1141	. . 3 |- (A e. On -> {x e. On | x ~~ (aleph` A)} = ((aleph` suc A) \ (aleph` A)))
15		fvex 2838	. . . . 5 |- (aleph` suc A) e. V
16		fvex 2838	. . . . 5 |- (aleph` A) e. V
17	15, 16	infdif 4948	. . . 4 |- ((om ~<_ (aleph` suc A) /\ (aleph` A) ~< (aleph` suc A)) -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
18		sucelon 2319	. . . . . 6 |- (A e. On <-> suc A e. On)
19		alephgeom 3687	. . . . . 6 |- (suc A e. On <-> om (_ (aleph` suc A))
20	18, 19	bitr 151	. . . . 5 |- (A e. On <-> om (_ (aleph` suc A))
21		ssdom2g 3312	. . . . . 6 |- ((aleph` suc A) e. V -> (om (_ (aleph` suc A) -> om ~<_ (aleph` suc A)))
22	15, 21	ax-mp 6	. . . . 5 |- (om (_ (aleph` suc A) -> om ~<_ (aleph` suc A))
23	20, 22	sylbi 174	. . . 4 |- (A e. On -> om ~<_ (aleph` suc A))
24		alephordlem1 3677	. . . 4 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
25	17, 23, 24	sylanc 361	. . 3 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
26	14, 25	eqbrtrd 2077	. 2 |- (A e. On -> {x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A))
27	15	ensym 3317	. 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 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   \ cdif 1484   (_ wss 1487   class class class wbr 2054  Oncon0 2199  suc csuc 2201  omcom 2372  ` cfv 2422   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273  cardccrd 3620  alephcale 3621

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-2o 3105  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624  df-cda 3715  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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