Theorem alephsuc3 7587

Description: An alternate representation of a successor aleph. Compare alephsuc 4877 and alephsuc2 4892. Equality can be obtained by taking the card of the right-hand side then using alephcard 4878 and carden 4841.

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 4892	. . . . . 6 |- (A e. On -> (aleph` suc A) = {x e. On | x ~<_ (aleph` A)})
2	1	difeq1d 2161	. . . . 5 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ (aleph` A)))
3		alephcard 4878	. . . . . . . 8 |- (card` (aleph` A)) = (aleph` A)
4		cardval2 4866	. . . . . . . 8 |- (card` (aleph` A)) = {x e. On | x ~< (aleph` A)}
5	3, 4	eqtr3 1500	. . . . . . 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 2162	. . . . 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 1510	. . . 4 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) = ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}))
9		difrab 2276	. . . . 5 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
10		bren2 4395	. . . . . . 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 1808	. . . . 5 |- {x e. On | x ~~ (aleph` A)} = {x e. On | (x ~<_ (aleph` A) /\ -. x ~< (aleph` A))}
13	9, 12	eqtr4 1501	. . . 4 |- ({x e. On | x ~<_ (aleph` A)} \ {x e. On | x ~< (aleph` A)}) = {x e. On | x ~~ (aleph` A)}
14	8, 13	syl6req 1527	. . 3 |- (A e. On -> {x e. On | x ~~ (aleph` A)} = ((aleph` suc A) \ (aleph` A)))
15		fvex 3738	. . . . 5 |- (aleph` suc A) e. V
16		fvex 3738	. . . . 5 |- (aleph` A) e. V
17	15, 16	infdif 7569	. . . 4 |- ((om ~<_ (aleph` suc A) /\ (aleph` A) ~< (aleph` suc A)) -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
18		sucelon 3074	. . . . . 6 |- (A e. On <-> suc A e. On)
19		alephgeom 4893	. . . . . 6 |- (suc A e. On <-> om (_ (aleph` suc A))
20	18, 19	bitr 173	. . . . 5 |- (A e. On <-> om (_ (aleph` suc A))
21		ssdom2g 4415	. . . . . 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 4883	. . . 4 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
25	17, 23, 24	sylanc 473	. . 3 |- (A e. On -> ((aleph` suc A) \ (aleph` A)) ~~ (aleph` suc A))
26	14, 25	eqbrtrd 2640	. 2 |- (A e. On -> {x e. On | x ~~ (aleph` A)} ~~ (aleph` suc A))
27	15	ensym 4418	. 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 958   e. wcel 960  {crab 1651  Vcvv 1814   \ cdif 2047   (_ wss 2050   class class class wbr 2624  Oncon0 2954  suc csuc 2956  omcom 3137  ` cfv 3188   ~~ cen 4370   ~<_ cdom 4371   ~< csdm 4372  cardccrd 4823  alephcale 4824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-iso 3205  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-2o 4140  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-fin 4377  df-card 4826  df-aleph 4827  df-cda 4930  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570

Copyright terms: Public domain