Theorem ondomon 9385

Description: The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 8449. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ondomon	|- ( A e. V -> { x e. On | x ~<_ A } e. On )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem ondomon

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

Step	Hyp	Ref	Expression
1		onelon 5748	. . . . . . . . . . . 12 |- ( ( z e. On /\ y e. z ) -> y e. On )
2		vex 3203	. . . . . . . . . . . . 13 |- z e. _V
3		onelss 5766	. . . . . . . . . . . . . 14 |- ( z e. On -> ( y e. z -> y C_ z ) )
4	3	imp 445	. . . . . . . . . . . . 13 |- ( ( z e. On /\ y e. z ) -> y C_ z )
5		ssdomg 8001	. . . . . . . . . . . . 13 |- ( z e. _V -> ( y C_ z -> y ~<_ z ) )
6	2, 4, 5	mpsyl 68	. . . . . . . . . . . 12 |- ( ( z e. On /\ y e. z ) -> y ~<_ z )
7	1, 6	jca 554	. . . . . . . . . . 11 |- ( ( z e. On /\ y e. z ) -> ( y e. On /\ y ~<_ z ) )
8		domtr 8009	. . . . . . . . . . . . 13 |- ( ( y ~<_ z /\ z ~<_ A ) -> y ~<_ A )
9	8	anim2i 593	. . . . . . . . . . . 12 |- ( ( y e. On /\ ( y ~<_ z /\ z ~<_ A ) ) -> ( y e. On /\ y ~<_ A ) )
10	9	anassrs 680	. . . . . . . . . . 11 |- ( ( ( y e. On /\ y ~<_ z ) /\ z ~<_ A ) -> ( y e. On /\ y ~<_ A ) )
11	7, 10	sylan 488	. . . . . . . . . 10 |- ( ( ( z e. On /\ y e. z ) /\ z ~<_ A ) -> ( y e. On /\ y ~<_ A ) )
12	11	exp31 630	. . . . . . . . 9 |- ( z e. On -> ( y e. z -> ( z ~<_ A -> ( y e. On /\ y ~<_ A ) ) ) )
13	12	com12 32	. . . . . . . 8 |- ( y e. z -> ( z e. On -> ( z ~<_ A -> ( y e. On /\ y ~<_ A ) ) ) )
14	13	impd 447	. . . . . . 7 |- ( y e. z -> ( ( z e. On /\ z ~<_ A ) -> ( y e. On /\ y ~<_ A ) ) )
15		breq1 4656	. . . . . . . 8 |- ( x = z -> ( x ~<_ A <-> z ~<_ A ) )
16	15	elrab 3363	. . . . . . 7 |- ( z e. { x e. On | x ~<_ A } <-> ( z e. On /\ z ~<_ A ) )
17		breq1 4656	. . . . . . . 8 |- ( x = y -> ( x ~<_ A <-> y ~<_ A ) )
18	17	elrab 3363	. . . . . . 7 |- ( y e. { x e. On | x ~<_ A } <-> ( y e. On /\ y ~<_ A ) )
19	14, 16, 18	3imtr4g 285	. . . . . 6 |- ( y e. z -> ( z e. { x e. On | x ~<_ A } -> y e. { x e. On | x ~<_ A } ) )
20	19	imp 445	. . . . 5 |- ( ( y e. z /\ z e. { x e. On | x ~<_ A } ) -> y e. { x e. On | x ~<_ A } )
21	20	gen2 1723	. . . 4 |- A. y A. z ( ( y e. z /\ z e. { x e. On | x ~<_ A } ) -> y e. { x e. On | x ~<_ A } )
22		dftr2 4754	. . . 4 |- ( 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 221	. . 3 |- Tr { x e. On | x ~<_ A }
24		ssrab2 3687	. . 3 |- { x e. On | x ~<_ A } C_ On
25		ordon 6982	. . 3 |- Ord On
26		trssord 5740	. . 3 |- ( ( 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 1424	. 2 |- Ord { x e. On | x ~<_ A }
28		elex 3212	. . . . . 6 |- ( A e. V -> A e. _V )
29		canth2g 8114	. . . . . . . . 9 |- ( A e. _V -> A ~< ~P A )
30		domsdomtr 8095	. . . . . . . . 9 |- ( ( x ~<_ A /\ A ~< ~P A ) -> x ~< ~P A )
31	29, 30	sylan2 491	. . . . . . . 8 |- ( ( x ~<_ A /\ A e. _V ) -> x ~< ~P A )
32	31	expcom 451	. . . . . . 7 |- ( A e. _V -> ( x ~<_ A -> x ~< ~P A ) )
33	32	ralrimivw 2967	. . . . . 6 |- ( A e. _V -> A. x e. On ( x ~<_ A -> x ~< ~P A ) )
34	28, 33	syl 17	. . . . 5 |- ( A e. V -> A. x e. On ( x ~<_ A -> x ~< ~P A ) )
35		ss2rab 3678	. . . . 5 |- ( { x e. On | x ~<_ A } C_ { x e. On | x ~< ~P A } <-> A. x e. On ( x ~<_ A -> x ~< ~P A ) )
36	34, 35	sylibr 224	. . . 4 |- ( A e. V -> { x e. On | x ~<_ A } C_ { x e. On | x ~< ~P A } )
37		pwexg 4850	. . . . . 6 |- ( A e. V -> ~P A e. _V )
38		numth3 9292	. . . . . 6 |- ( ~P A e. _V -> ~P A e. dom card )
39		cardval2 8817	. . . . . 6 |- ( ~P A e. dom card -> ( card ` ~P A ) = { x e. On | x ~< ~P A } )
40	37, 38, 39	3syl 18	. . . . 5 |- ( A e. V -> ( card ` ~P A ) = { x e. On | x ~< ~P A } )
41		fvex 6201	. . . . 5 |- ( card ` ~P A ) e. _V
42	40, 41	syl6eqelr 2710	. . . 4 |- ( A e. V -> { x e. On | x ~< ~P A } e. _V )
43		ssexg 4804	. . . 4 |- ( ( { x e. On | x ~<_ A } C_ { x e. On | x ~< ~P A } /\ { x e. On | x ~< ~P A } e. _V ) -> { x e. On | x ~<_ A } e. _V )
44	36, 42, 43	syl2anc 693	. . 3 |- ( A e. V -> { x e. On | x ~<_ A } e. _V )
45		elong 5731	. . 3 |- ( { x e. On | x ~<_ A } e. _V -> ( { x e. On | x ~<_ A } e. On <-> Ord { x e. On | x ~<_ A } ) )
46	44, 45	syl 17	. 2 |- ( A e. V -> ( { x e. On | x ~<_ A } e. On <-> Ord { x e. On | x ~<_ A } ) )
47	27, 46	mpbiri 248	1 |- ( A e. V -> { x e. On | x ~<_ A } e. On )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   Tr wtr 4752   dom cdm 5114   Ord word 5722   Oncon0 5723   ` cfv 5888   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761

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-ac2 9285

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-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-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-ac 8939

This theorem is referenced by: (None)