Theorem alephsuc2 8903

Description: An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 8449 function by transfinite recursion, starting from om. Using this theorem we could define the aleph function with { z e. On | z ~<_ x } in place of |^| { z e. On | x ~< z } in df-aleph 8766. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
alephsuc2	|- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )

Distinct variable group:   x, A

Proof of Theorem alephsuc2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephsucdom 8902	. . 3 |- ( A e. On -> ( x ~<_ ( aleph ` A ) <-> x ~< ( aleph ` suc A ) ) )
2	1	rabbidv 3189	. 2 |- ( A e. On -> { x e. On | x ~<_ ( aleph ` A ) } = { x e. On | x ~< ( aleph ` suc A ) } )
3		alephon 8892	. . . . . . 7 |- ( aleph ` suc A ) e. On
4	3	oneli 5835	. . . . . 6 |- ( y e. ( aleph ` suc A ) -> y e. On )
5		alephcard 8893	. . . . . . . . 9 |- ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A )
6		iscard 8801	. . . . . . . . 9 |- ( ( card ` ( aleph ` suc A ) ) = ( aleph ` suc A ) <-> ( ( aleph ` suc A ) e. On /\ A. y e. ( aleph ` suc A ) y ~< ( aleph ` suc A ) ) )
7	5, 6	mpbi 220	. . . . . . . 8 |- ( ( aleph ` suc A ) e. On /\ A. y e. ( aleph ` suc A ) y ~< ( aleph ` suc A ) )
8	7	simpri 478	. . . . . . 7 |- A. y e. ( aleph ` suc A ) y ~< ( aleph ` suc A )
9	8	rspec 2931	. . . . . 6 |- ( y e. ( aleph ` suc A ) -> y ~< ( aleph ` suc A ) )
10	4, 9	jca 554	. . . . 5 |- ( y e. ( aleph ` suc A ) -> ( y e. On /\ y ~< ( aleph ` suc A ) ) )
11		sdomel 8107	. . . . . . 7 |- ( ( y e. On /\ ( aleph ` suc A ) e. On ) -> ( y ~< ( aleph ` suc A ) -> y e. ( aleph ` suc A ) ) )
12	3, 11	mpan2 707	. . . . . 6 |- ( y e. On -> ( y ~< ( aleph ` suc A ) -> y e. ( aleph ` suc A ) ) )
13	12	imp 445	. . . . 5 |- ( ( y e. On /\ y ~< ( aleph ` suc A ) ) -> y e. ( aleph ` suc A ) )
14	10, 13	impbii 199	. . . 4 |- ( y e. ( aleph ` suc A ) <-> ( y e. On /\ y ~< ( aleph ` suc A ) ) )
15		breq1 4656	. . . . 5 |- ( x = y -> ( x ~< ( aleph ` suc A ) <-> y ~< ( aleph ` suc A ) ) )
16	15	elrab 3363	. . . 4 |- ( y e. { x e. On | x ~< ( aleph ` suc A ) } <-> ( y e. On /\ y ~< ( aleph ` suc A ) ) )
17	14, 16	bitr4i 267	. . 3 |- ( y e. ( aleph ` suc A ) <-> y e. { x e. On | x ~< ( aleph ` suc A ) } )
18	17	eqriv 2619	. 2 |- ( aleph ` suc A ) = { x e. On | x ~< ( aleph ` suc A ) }
19	2, 18	syl6reqr 2675	1 |- ( A e. On -> ( aleph ` suc A ) = { x e. On | x ~<_ ( aleph ` A ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   Oncon0 5723   suc csuc 5725   ` cfv 5888   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephsuc3  9402