Theorem card2on 7523

Description: Proof that the alternate definition cardval2 7879 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)

Assertion

Ref	Expression
card2on	|- { x e. On | x ~< A } e. On

Distinct variable group:   x, A

Proof of Theorem card2on

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelon 4607	. . . . . . . . . . . . 13 |- ( ( z e. On /\ y e. z ) -> y e. On )
2		vex 2960	. . . . . . . . . . . . . 14 |- z e. _V
3		onelss 4624	. . . . . . . . . . . . . . 15 |- ( z e. On -> ( y e. z -> y C_ z ) )
4	3	imp 420	. . . . . . . . . . . . . 14 |- ( ( z e. On /\ y e. z ) -> y C_ z )
5		ssdomg 7154	. . . . . . . . . . . . . 14 |- ( z e. _V -> ( y C_ z -> y ~<_ z ) )
6	2, 4, 5	mpsyl 62	. . . . . . . . . . . . 13 |- ( ( z e. On /\ y e. z ) -> y ~<_ z )
7	1, 6	jca 520	. . . . . . . . . . . 12 |- ( ( z e. On /\ y e. z ) -> ( y e. On /\ y ~<_ z ) )
8		domsdomtr 7243	. . . . . . . . . . . . . 14 |- ( ( y ~<_ z /\ z ~< A ) -> y ~< A )
9	8	anim2i 554	. . . . . . . . . . . . 13 |- ( ( y e. On /\ ( y ~<_ z /\ z ~< A ) ) -> ( y e. On /\ y ~< A ) )
10	9	anassrs 631	. . . . . . . . . . . 12 |- ( ( ( y e. On /\ y ~<_ z ) /\ z ~< A ) -> ( y e. On /\ y ~< A ) )
11	7, 10	sylan 459	. . . . . . . . . . 11 |- ( ( ( z e. On /\ y e. z ) /\ z ~< A ) -> ( y e. On /\ y ~< A ) )
12	11	exp31 589	. . . . . . . . . 10 |- ( z e. On -> ( y e. z -> ( z ~< A -> ( y e. On /\ y ~< A ) ) ) )
13	12	com12 30	. . . . . . . . 9 |- ( y e. z -> ( z e. On -> ( z ~< A -> ( y e. On /\ y ~< A ) ) ) )
14	13	imp3a 422	. . . . . . . 8 |- ( y e. z -> ( ( z e. On /\ z ~< A ) -> ( y e. On /\ y ~< A ) ) )
15		breq1 4216	. . . . . . . . 9 |- ( x = z -> ( x ~< A <-> z ~< A ) )
16	15	elrab 3093	. . . . . . . 8 |- ( z e. { x e. On | x ~< A } <-> ( z e. On /\ z ~< A ) )
17		breq1 4216	. . . . . . . . 9 |- ( x = y -> ( x ~< A <-> y ~< A ) )
18	17	elrab 3093	. . . . . . . 8 |- ( y e. { x e. On | x ~< A } <-> ( y e. On /\ y ~< A ) )
19	14, 16, 18	3imtr4g 263	. . . . . . 7 |- ( y e. z -> ( z e. { x e. On | x ~< A } -> y e. { x e. On | x ~< A } ) )
20	19	imp 420	. . . . . 6 |- ( ( y e. z /\ z e. { x e. On | x ~< A } ) -> y e. { x e. On | x ~< A } )
21	20	gen2 1557	. . . . 5 |- A. y A. z ( ( y e. z /\ z e. { x e. On | x ~< A } ) -> y e. { x e. On | x ~< A } )
22		dftr2 4305	. . . . 5 |- ( Tr { x e. On | x ~< A } <-> A. y A. z ( ( y e. z /\ z e. { x e. On | x ~< A } ) -> y e. { x e. On | x ~< A } ) )
23	21, 22	mpbir 202	. . . 4 |- Tr { x e. On | x ~< A }
24		ssrab2 3429	. . . 4 |- { x e. On | x ~< A } C_ On
25		ordon 4764	. . . 4 |- Ord On
26		trssord 4599	. . . 4 |- ( ( Tr { x e. On | x ~< A } /\ { x e. On | x ~< A } C_ On /\ Ord On ) -> Ord { x e. On | x ~< A } )
27	23, 24, 25, 26	mp3an 1280	. . 3 |- Ord { x e. On | x ~< A }
28		hartogs 7514	. . . 4 |- ( A e. _V -> { x e. On | x ~<_ A } e. On )
29		sdomdom 7136	. . . . . . 7 |- ( x ~< A -> x ~<_ A )
30	29	a1i 11	. . . . . 6 |- ( x e. On -> ( x ~< A -> x ~<_ A ) )
31	30	ss2rabi 3426	. . . . 5 |- { x e. On | x ~< A } C_ { x e. On | x ~<_ A }
32		ssexg 4350	. . . . 5 |- ( ( { x e. On | x ~< A } C_ { x e. On | x ~<_ A } /\ { x e. On | x ~<_ A } e. On ) -> { x e. On | x ~< A } e. _V )
33	31, 32	mpan 653	. . . 4 |- ( { x e. On | x ~<_ A } e. On -> { x e. On | x ~< A } e. _V )
34		elong 4590	. . . 4 |- ( { x e. On | x ~< A } e. _V -> ( { x e. On | x ~< A } e. On <-> Ord { x e. On | x ~< A } ) )
35	28, 33, 34	3syl 19	. . 3 |- ( A e. _V -> ( { x e. On | x ~< A } e. On <-> Ord { x e. On | x ~< A } ) )
36	27, 35	mpbiri 226	. 2 |- ( A e. _V -> { x e. On | x ~< A } e. On )
37		0elon 4635	. . . 4 |- (/) e. On
38		eleq1 2497	. . . 4 |- ( { x e. On | x ~< A } = (/) -> ( { x e. On | x ~< A } e. On <-> (/) e. On ) )
39	37, 38	mpbiri 226	. . 3 |- ( { x e. On | x ~< A } = (/) -> { x e. On | x ~< A } e. On )
40		df-ne 2602	. . . . 5 |- ( { x e. On | x ~< A } =/= (/) <-> -. { x e. On | x ~< A } = (/) )
41		rabn0 3648	. . . . 5 |- ( { x e. On | x ~< A } =/= (/) <-> E. x e. On x ~< A )
42	40, 41	bitr3i 244	. . . 4 |- ( -. { x e. On | x ~< A } = (/) <-> E. x e. On x ~< A )
43		relsdom 7117	. . . . . 6 |- Rel ~<
44	43	brrelex2i 4920	. . . . 5 |- ( x ~< A -> A e. _V )
45	44	rexlimivw 2827	. . . 4 |- ( E. x e. On x ~< A -> A e. _V )
46	42, 45	sylbi 189	. . 3 |- ( -. { x e. On | x ~< A } = (/) -> A e. _V )
47	39, 46	nsyl4 137	. 2 |- ( -. A e. _V -> { x e. On | x ~< A } e. On )
48	36, 47	pm2.61i 159	1 |- { x e. On | x ~< A } e. On

Syntax hints:   -. wn 3   -> wi 4   <-> wb 178   /\ wa 360   A.wal 1550   = wceq 1653   e. wcel 1726   =/= wne 2600   E.wrex 2707   {crab 2710   _Vcvv 2957   C_ wss 3321   (/)c0 3629   class class class wbr 4213   Tr wtr 4303   Ord word 4581   Oncon0 4582   ~<_ cdom 7108   ~< csdm 7109

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-riota 6550  df-recs 6634  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-oi 7480