Theorem ssonprc 6992

Description: Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)

Assertion

Ref	Expression
ssonprc	|- ( A C_ On -> ( A e/ _V <-> U. A = On ) )

Proof of Theorem ssonprc

Step	Hyp	Ref	Expression
1		df-nel 2898	. 2 |- ( A e/ _V <-> -. A e. _V )
2		ssorduni 6985	. . . . . . . 8 |- ( A C_ On -> Ord U. A )
3		ordeleqon 6988	. . . . . . . 8 |- ( Ord U. A <-> ( U. A e. On \/ U. A = On ) )
4	2, 3	sylib 208	. . . . . . 7 |- ( A C_ On -> ( U. A e. On \/ U. A = On ) )
5	4	orcomd 403	. . . . . 6 |- ( A C_ On -> ( U. A = On \/ U. A e. On ) )
6	5	ord 392	. . . . 5 |- ( A C_ On -> ( -. U. A = On -> U. A e. On ) )
7		uniexr 6972	. . . . 5 |- ( U. A e. On -> A e. _V )
8	6, 7	syl6 35	. . . 4 |- ( A C_ On -> ( -. U. A = On -> A e. _V ) )
9	8	con1d 139	. . 3 |- ( A C_ On -> ( -. A e. _V -> U. A = On ) )
10		onprc 6984	. . . 4 |- -. On e. _V
11		uniexg 6955	. . . . 5 |- ( A e. _V -> U. A e. _V )
12		eleq1 2689	. . . . 5 |- ( U. A = On -> ( U. A e. _V <-> On e. _V ) )
13	11, 12	syl5ib 234	. . . 4 |- ( U. A = On -> ( A e. _V -> On e. _V ) )
14	10, 13	mtoi 190	. . 3 |- ( U. A = On -> -. A e. _V )
15	9, 14	impbid1 215	. 2 |- ( A C_ On -> ( -. A e. _V <-> U. A = On ) )
16	1, 15	syl5bb 272	1 |- ( A C_ On -> ( A e/ _V <-> U. A = On ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   e/ wnel 2897   _Vcvv 3200   C_ wss 3574   U.cuni 4436   Ord word 5722   Oncon0 5723

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  inaprc  9658