Theorem alephsuc3 7585

Description: An alternate representation of a successor aleph. Compare alephsuc 4866 and alephsuc2 4881. Equality can be obtained by taking the card of the right-hand side then using alephcard 4867 and carden 4831.

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 4881	. . . . . 6 |- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})
2	1	difeq1d 2158	. . . . 5 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ (aleph` A)))
3		alephcard 4867	. . . . . . . 8 |- (card` (aleph` A)) = (aleph` A)
4		cardval2 4855	. . . . . . . 8 |- (card` (aleph` A)) = {x e. On | x ~< (aleph` A)}
5	3, 4	eqtr3 1497	. . . . . . 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 2159	. . . . 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 1507	. . . 4 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}))
9		difrab 2273	. . . . 5 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
10		bren2 4389	. . . . . . 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	rabbii 1805	. . . . 5 |- {x e. On | x ~~ (aleph` A)} = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
13	9, 12	eqtr4 1498	. . . 4 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | x ~~ (aleph` A)}
14	8, 13	syl6req 1524	. . 3 |- (A e. On -> {x e. On | x ~~ (aleph` A)} = ((aleph` suc A) \ (aleph` A)))
15		fvex 3732	. . . . 5 |- (aleph` suc A) e. V
16		fvex 3732	. . . . 5 |- (aleph` A) e. V
17	15, 16	infdif 7568	. . . 4 |- ((om ~<_ (aleph` suc A) /\ (aleph` A) ~< (aleph` suc A)) -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
18		sucelon 3068	. . . . . 6 |- (A e. On <-> suc A e. On)
19		alephgeom 4882	. . . . . 6 |- (suc A e. On <-> om (_ (aleph` suc A))
20	18, 19	bitr 173	. . . . 5 |- (A e. On <-> om (_ (aleph` suc A))
21		ssdom2g 4409	. . . . . 6 |- ((aleph` suc A) e. V -> (om (_ (aleph` suc A) -> om ~<_ (aleph` suc A)))
22	15, 21	ax-mp 7	. . . . 5 |- (om (_ (aleph` suc A) -> om ~<_ (aleph` suc A))
23	20, 22	sylbi 199	. . . 4 |- (A e. On -> om ~<_ (aleph` suc A))
24		alephordlem1 4872	. . . 4 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
25	17, 23, 24	sylanc 471	. . 3 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
26	14, 25	eqbrtrd 2635	. 2 |- (A e. On -> {x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A))
27	15	ensym 4412	. 2 |- ({x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A) -> (aleph` suc A) ~~ {x e. On | x ~~ (aleph` A)})
28	26, 27	syl 10	1 |- (A e. On -> (aleph` suc A) ~~ {x e. On | x ~~ (aleph` A)})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   \ cdif 2044   (_ wss 2047   class class class wbr 2619  Oncon0 2948  suc csuc 2950  omcom 3131  ` cfv 3182   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366  cardccrd 4813  alephcale 4814

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-iso 3199  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-2o 4134  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817  df-cda 4918  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-n 5925  df-2 5970  df-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569

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